Advances in Differential Equations
Volume 18, Numbers 1-2 (2013), 1–48
EXISTENCE THEOREMS FOR QUASILINEAR ELLIPTIC EIGENVALUE PROBLEMS IN UNBOUNDED DOMAINS Giuseppina Autuori and Patrizia Pucci Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia Via Vanvitelli 1, 06123 Perugia, Italy Csaba Varga Faculty of Mathematics and Computer Science, Babe¸s–Bolyai University 400084 Cluj–Napoca, Romania (Submitted by: Reza Aftabizadeh) Abstract. The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter λ under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted pLaplacian operator and subcritical nonlinearities, and even in the case p = 2 the main existence results are new. Denoting by λ1 the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case λ ≥ λ1 or λ < λ1 . In the first part of the paper we show the existence of a nontrivial solution for all λ ∈ R for the problem under Ambrosetti– Rabinowitz-type conditions on the nonlinearities involved in the model. In detail, we apply the mountain-pass theorem of Ambrosetti and Rabinowitz if λ < λ1 , while we use mini-max methods and linking structures over cones, as in Degiovanni [10] and in Degiovanni and Lancelotti [11], if λ ≥ λ1 . In the latter part of the paper we do not require any longer the Ambrosetti–Rabinowitz condition at ∞, but the so-called Szulkin– Weth conditions, and we obtain the same result for all λ ∈ R. More precisely, using the Nehari-manifold method for C 1 functionals developed by Szulkin and Weth in [38], we prove existence of ground states, multiple solutions, and least-energy sign-changing solutions, whenever λ < λ1 . On the other hand, in the case λ ≥ λ1 , we establish the existence of solutions again by linking methods.
1. Introduction In this paper we are concerned with problems arising in the study of physical phenomena related to the equilibrium of anisotropic continuous media Accepted for publication: August 2012. AMS Subject Classifications: 35J66, 35J20; 35J25, 35J70. 1
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
which possibly are somewhere perfect insulators; cf. Dautray and Lions [9] and also [37]. More precisely, let Ω ⊂ RN be an unbounded domain with (possibly noncompact) smooth boundary ∂Ω and take 1 < p < N . Consider the problem ( − div(a(x)|∇u|p−2 ∇u) = λf (x)|u|p−2 u + g(x, u) in Ω, (1.1) p−2 p−2 a(x)|∇u| ∇u · ν + b(x)|u| u = h(x, u) on ∂Ω, where λ is a real parameter and ν denotes the unit outward normal on ∂Ω. The main result of the paper consists in proving the existence of a nontrivial weak solution u of (1.1) for all λ ∈ R, requiring on the nonlinearities g and h either some Ambrosetti–Rabinowitz assumptions at ∞ or the so-called Szulkin–Weth conditions. The Robin boundary condition in (1.1) arises naturally in heat conduction problems as well as in physical geodesy; cf. [22]. Problem (1.1) may also be viewed as a prototype in the study of activator-inhibitor systems modeling biological pattern formation, in addition to a steady-state problem for a chemotactic aggregation model. For a more exhaustive physical discussion on the topic we refer to [19]. Problems of type (1.1) were studied recently in [19]–[21] and [25]–[27]. In these papers the authors use mountain-pass-type theorems, the fibering method due to Pohozaev and Ricceri’s critical-point-type theorems in order to prove the existence and multiplicity of solutions. Most of these papers deal with the case in which Ω is a bounded domain. The loss of compactness of the Sobolev imbeddings on unbounded domains renders variational techniques more delicate. Some of the papers treating problems on unbounded domains use special function spaces where the compactness is preserved, such as spaces of radially symmetric functions. We point out that even if Ω is unbounded, standard compact imbeddings still remain true, e.g., if Ω is thin at infinity, in the sense that lim sup meas(Ω ∩ B(x, 1)) : x ∈ RN , |x| = R = 0, R→∞
where meas denotes the Lebesgue measure and B(x, 1) is the unit ball centered at x. Such arguments cannot be applied to the general unbounded domains Ω we consider in this paper. Indeed, since Ω is not necessarily “thin” and it may look like RN at infinity (e.g., when Ω is an exterior domain), the analysis of the compactness failure shows that a Palais–Smale sequence, briefly (P S) sequence, of the associated energy functional (see
Existence theorems for eigenvalue problems
3
Bahri and Lions [5]) differs from its weak limit by “waves” that go to infinity. However, the definition of an appropriate solution space E, defined in Section 2, combined with the main assumptions ensures that E is compactly embedded into the weighted Lebesgue spaces involved. In the first part of this paper, using the method developed by Degiovanni and Lancelotti in [11], Degiovanni in [10], and the Ambrosetti and Rabinowitz mountain-pass theorem, we prove that (1.1) has a nontrivial solution for all λ ∈ R, when g and h satisfy an Ambrosetti–Rabinowitz -type condition. In the latter part we complete the picture, proving the existence of nontrivial solutions of (1.1) for all λ ∈ R, without requiring the famous Ambrosetti and Rabinowitz condition on the nonlinearities g and h, but rather the so-called Szulkin–Weth conditions. In [11] Degiovanni and Lancelotti prove the existence of nontrivial solutions for the equation given in (1.1) for any λ ∈ R, when a ≡ 1 and b = h ≡ 0, that is, under homogeneous Dirichlet boundary conditions on bounded domains Ω. They use a mini-max approach and construct linking structures over cones for an associated eigenvalue problem. In the case p 6= 2, even when all the weights involved in (1.1) are just trivial constants, a = f ≡ 1 and b = h ≡ 0, and Ω is a bounded domain of RN , the spectrum of −∆p with homogeneous Dirichlet boundary conditions is not at all clear. We refer to [11] and the references therein for a complete discussion on the complexity of this question, as well as on the difficulty in covering the case λ ≥ λ1 , where λ1 is the first eigenvalue of the natural underlying eigenvalue problem. For the above reasons, if λ ≥ λ1 , we prove existence of nontrivial solutions of (1.1) by constructing a suitable non-decreasing sequence (λk )k diverging to ∞ as k → ∞, via a mini-max argument based on a Z2 -cohomological index of the most canonical Finsler manifold M associated to the eigenvalue problem related to (1.1); see (3.1) below. The case λ < λ1 is either simply proved via the mountain-pass theorem of Ambrosetti and Rabinowitz [3] (see also [27] for similar problems), or using the Nehari manifold method for C 1 functionals developed by Szulkin and Weth in [38]. The paper is organized as follows. In Section 2 we present notation and some auxiliary results, some of them crucial to constructing the main sequence (λk )k used in the main proofs of Theorems 4.3 and 7.4. In Section 3 a simple minimization argument shows the existence of a positive first eigenvalue λ1 of the Robin nonlinear boundary eigenvalue problem corresponding to (1.1). The main properties of λ1 are briefly presented in Propositions 3.1 and 3.3. In Section 4 we prove the existence Theorem 4.3
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
under the Ambrosetti–Rabinowitz condition on g and h. In Section 5, we describe the abstract Nehari method and prove results of independent interest. From Section 6 on we assume the Szulkin–Weth conditions on g and h and prove several prelimary results of the main functionals associated to (1.1) in Section 6. In Section 7 we first show when λ < λ1 in Theorem 7.1 that problem (1.1) has a ground-state solution, that is, a nontrivial solution of minimal energy; that (1.1) admits a least-energy sign-changing solution; and that (1.1) has infinitely many pairs of solutions if furthermore g and h are assumed to be odd in u. In the final part of Section 7, when λ ≥ λ1 , we prove in Theorem 7.4 that (1.1) admits a nontrivial solution. The last part of the paper contains an Appendix in which we show that E = (E, k · k) is uniformly convex, where k · k is an appropriate norm in the solution space E. 2. Preliminaries and auxiliary results In this section we collect a series of notations and preliminaries which are used throughout the paper. By w we denote a weight on Ω, that is, a measurable function with w > 0 a.e. in Ω, so that Lσ (Ω, w), σ ≥ 1, is the weighted Lebesgue space equipped with the norm Z 1/σ kukσ,w = w(x)|u(x)|σ dx . Ω
Similarly, if w ˜ is a weight on ∂Ω, that is, w ˜ is measurable and a.e. positive in ∂Ω with respect to the N − 1-dimensional measure on ∂Ω, then Lσ (∂Ω, w), ˜ σ ≥ 1, denotes the weighted Lebesgue space equipped with the norm Z 1/σ σ kukσ,w,∂Ω = w(x)|u(x)| ˜ dS . ˜ ∂Ω
The next lemma, stated here for the weighted space Lσ (A, w), where A is a measurable subset of Rd , d ≥ 1, w is a weight on A, is well-known in the usual Lebesgue spaces (see, for instance, Theorem 4.9 of [7]). The proof is left to the reader, since it is standard; see also [34]. Lemma 2.1. If (un )n and u are in Lα (A, w), with α ∈ [1, ∞), and un → u in Lα (A, w) as n → ∞, then there exist a subsequence (unk )k of (un )n and a function h ∈ Lα (A, w) such that a.e. in Ω (i)
unk → u as k → ∞;
(ii)
|unk (x)| ≤ h(x)
for all k ∈ N.
The case (i) of the result below is proved for the standard Lebesgue spaces in Theorem 2.3 of [13], while for (ii) we refer to Lemma 4.2. of [11].
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Lemma 2.2. Let α, β ∈ [1, ∞), ϕ : A × R → R be a Carath´eodory function and let Nϕ (u) = ϕ(x, u) be the corresponding Nemytskii operator. (i) If |ϕ(x, s)| ≤ ϕ0 (x) + c|s|α/β for a.a. x ∈ A and all s ∈ R, where ϕ0 ∈ Lβ (A, w) and c > 0 is a constant, then Nϕ is continuous and bounded from Lα (A, w) into Lβ (A, w). (ii) Assume that for every ε > 0 there exists a nonnegative function aε ∈ Lβ (A, w) such that |ϕ(x, s)| ≤ aε (x) + ε|s|α/β for a.a. x ∈ A and all s ∈ R. If (un )n is a bounded sequence in Lα (A, w) and un → u a.e. in A, then Nϕ (un ) → Nϕ (u) in Lβ (A, w). Proof. (i) From kNϕ (u)kβ,w ≤ 2β−1 max{1, c} kϕ0 kββ,w + kukαα,w , it follows at once that Nϕ is well defined and bounded from Lα (A, w) into Lβ (A, w). Now, let (un )n ⊂ Lα (A, w) and u ∈ Lα (A, w) be such that un → u in α L (A, w). Fix a subsequence (unk )k of (un )n . By Lemma 2.1 there exists a subsequence, say, (vnk )k of (unk )k such that vnk → u and |vnk | ≤ h a.e. in A, for some appropriate h ∈ Lα (A, w). Hence Nϕ (vnk ) → Nϕ (u) a.e. in A and |Nϕ (vnk )| ≤ ϕ0 + chα/β ∈ Lβ (A, w). Therefore, by the Lebesgue dominated convergence theorem we get that Nϕ (vnk ) → Nϕ (u) in Lβ (A, w). Thus, the entire sequence (Nϕ (un ))n converges to Nϕ (u) in Lβ (A, w) as n → ∞. This proves the continuity of Nϕ . (ii) Clearly, for a.a. x ∈ A and all n ∈ N we have |ϕ(x, un (x)) − ϕ(x, u(x))| ≤ 4β−1 {2aε (x)β + εβ |un (x)|α + εβ |u(x)|α }. By the Fatou lemma n o 2 · 4β−1 kaε kββ,w + εβ kukαα,w ≤ 2 · 4β−1 kaε kββ,w n o + 4β−1 εβ kukαα,w + sup kun kαα,w − lim sup kϕ(x, un ) − ϕ(x, u)kββ,w . n
n
Therefore, n o 0 ≤ lim sup kϕ(x, un ) − ϕ(x, u)kββ,w ≤ 4β−1 εβ sup kun kαα,w − kukαα,w . n
n
By the arbitrariness of ε > 0, we get the assertion.
Let Cδ∞ (Ω) be the space of functions of class C0∞ (RN ) restricted on Ω and let E be the completion of Cδ∞ (Ω) with respect to the norm Z |u(x)|p 1/p dx . kukE = |∇u(x)|p + (1 + |x|)p Ω
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
Of course, since Ω is a smooth domain, by the celebrated density Theorem 3.18 of Adams [1] it is apparent that E = W 1,p (Ω) and k · kE is an equivalent norm on W 1,p (Ω), whenever Ω is bounded. On the other hand, 1,p if Ω is unbounded, then E ⊂ Wloc (Ω). We are now able to state the main result of the paper, which is proved in Section 5. Throughout the paper, without further mentioning it, we assume that 0 < a0 ≤ a ∈ C 1 (Ω) ∩ L∞ (Ω), b : ∂Ω → R is a continuous function, with cb (1 + |x|)1−p ≤ b(x) ≤ Cb (1 + |x|)1−p , for some constants 0 < cb ≤ Cb ; moreover, f is a measurable weight defined on Ω and satisfying 0 < f (x) ≤ Cf w1 (x),
w1 (x) = (1 + |x|)−α1 ,
p < α1 < N,
(2.1)
for a.a. x ∈ Ω. Since the embedding E ,→ Lp (∂Ω, b) is continuous by Theorem 1 of [31], for all u ∈ E the quantity Z Z 1/p p kuk = a(x)|∇u(x)| dx + b(x)|u(x)|p dS Ω
∂Ω
is well defined and k · k is an equivalent norm in E, as shown in [31, Lemma 2]. From now on we endow E with the norm k · k. Proposition A.2 of the Appendix shows that (E, k · k) is a uniformly convex Banach space. From now on E ? = (E ? , k · kE ? ) denotes the dual space of E = (E, k · k). Proposition 2.3. The map F : E → E ? , defined by F(u) = f |u|p−2 u, is weak-to-strong sequentially continuous; i.e., un * u in E implies kF(un ) − F(u)kE ? → 0 as n → ∞. Let w be a weight on Ω such that the embedding E ,→ Lp (Ω, w) is compact. Then, also I : E → L1 (Ω), defined by I(u) = w|u|p , is weak-to-strong sequentially continuous; that is, un * u in E implies kI(un ) − I(u)k1 → 0 as n → ∞. Similarly, if w ˜ be a weight on ∂Ω such that the embedding E ,→ Lp (∂Ω, w) ˜ ˜ is compact, then I˜ : E → L1 (∂Ω), I(u) = w|u| ˜ p , is weak-to-strong sequen˜ n ) − I(u)k ˜ tially continuous; that is, un * u in E implies kI(u 1,∂Ω → 0 as n → ∞. Proof. Let (un )n ⊂ E be such that un * u in E. Hence un → u in Lp (Ω, f ), since the embedding E ,→ Lp (Ω, f ) is compact, because α1 ∈ (p, N ). Thus, in particular, there exists C = C(p, N, Ω) > 0 such that kvkp,f ≤ Ckvk
for all v ∈ E.
(2.2)
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Clearly, kun kp,f → kukp,f , that is kvn kp0 ,f → kvkp0 ,f , where vn = |un |p−2 un 0 and v = |u|p−2 u. We claim that vn → v in Lp (Ω, f ). Indeed, fix any subsequence (vnk )k of (vn )n . The related subsequence (unk )k of (un )n converges in Lp (Ω, f ) and admits a subsequence, say (unkj )j , converging to u a.e. in Ω by Lemma 2.1. Hence, the corresponding subsequence (vnkj )j of (vnk )k converges to v a.e. in Ω. Therefore, since 1 < p0 < ∞, by the Clarkson and 0 Mil’man theorems it follows that vnkj * v in Lp (Ω, f ), since the sequence (kvn kp0 ,f )n is bounded, and so by Radon’s theorem we get that vnkj → v in 0
Lp (Ω, f ), since kvn kp0 ,f → kvkp0 ,f . This shows the claim, since the subsequence (vnk )k of (vn )n is arbitrary. By H¨ older’s inequality we have for all φ ∈ E, with kφk = 1, Z |hF(un )−F(u), φi| ≤ f (x)|vn −v|·|φ| dx ≤ kvn −vkp0,f kφkp,f ≤ Ckvn −vkp0,f , Ω
where C > 0 is given in (2.2). In other words, kF(un ) − F(u)kE ? → 0 as n → ∞. Since E ,→ Lp (Ω, w) and E ,→ Lp (∂Ω, w) ˜ are compact by assumption, the second and the third parts of the statement are trivial. Indeed, by the elementary inequality ||s|p − |t|p | ≤ Cp (|s − t|p + |y|p−1 |s − t|) for all s, t ∈ R, where Cp is an appropriate constant depending only on p > 1, we find Z Z |I(un ) − I(u)| dx = w(x)||un |p − |u|p | dx Ω Ω Z ≤ Cp w(x) |un − u|p + |u|p−1 |un − u| dx Ω ≤ Cp kun − ukpp,w + kukp−1 p,w kun − ukp,w → 0, since un * u in E and so un → u in Lp (Ω, w) as n → ∞. ˜ Similarly, we get the claim for I.
In the case λ ≥ λ1 we are going to solve problem (1.1) with mini-max and linking arguments based on a Z2 -cohomological index. For this purpose let us present some crucial auxiliary results. Let X = (X, k · kX ) be a real normed space and X ? = (X ? , k · kX ? ) its dual. We denote by S(X) the set of all center-symmetric subsets of X not containing the origin of X. For A ∈ S(X), put A = A/Z2 . Let ϕ : A → RP ∞ be the classifying map and ϕ? : H ? (RP ∞ ) = Z2 [ω] → H ? (A) the induced homomorphism of the cohomology rings. The cohomological index of A, denoted by i(A), is
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
defined by sup{k ≥ 1 : ϕ? (ω k−1 ) 6= 0}. Let A, B ∈ S(X), and let us list some properties which will be used in the sequel. (i1 ) Monotonicity: if A ⊂ B, then i(A) ≤ i(B). (i2 ) Invariance: if ϕ : A → B is an odd homeomorphism, then i(A) = i(B). (i3 ) Continuity: if C is a closed symmetric subset of A, then there exists a closed symmetric neighborhood N of C in A, such that i(N ) = i(C); hence, the interior of N is also a neighborhood of C in A and i(int C) = i(C). (i4 ) Neighborhood of the origin: if U is a bounded closed symmetric neighborhood of the origin in X, then i(∂U ) = dim X. Definition 2.4. Let I : X → R be a functional of class C 1 (X). A sequence (un )n ⊂ X is said to be a Palais–Smale sequence for I, (P S) sequence for short, if (I(un ))n is bounded and kI0 (un )||X ? → 0 as n → ∞. The functional I satisfies the Palais–Smale condition, (P S) condition, if any (P S) sequence admits a convergent subsequence. Similarly, I satisfies the Palais–Smale condition at level c, (P S)c condition for short, if any (P S) sequence, with I(un ) → c as n → ∞, admits a convergent subsequence. Hence, equivalently, I satisfies the (P S) condition if and only if (P S)c holds for I at every level c ∈ R. We recall a well-known result which is crucial in the proof that problem (3.1) has infinitely many distinct eigenvalues which tend to infinity. Proposition 2.5 (Proposition 3.14.7 of [30]). Let M be a C 1 Finsler complete manifold with free Z2 -action, and let I ∈ C 1 (M) be even (i.e., Z2 invariant). Set Fk = {M ⊂ M : M is symmetric and i(M ) ≥ k} and λk = inf sup I(u). Then the following statements are true: M ∈Fk u∈M
(i) If −∞ < λk = · · · = λk+m−1 = c < ∞ and I satisfies (P S)c , then i(K c ) ≥ m, where K c = {u ∈ M : I(u) = c, I0 (u) = 0}. (ii) If −∞ < λk ≤ · · · ≤ λk+m−1 < ∞ and I satisfies (P S)c for c in {λk , . . . , λk+m−1 }, then all λk , . . . , λk+m−1 are critical values for I and I has m distinct pairs of associated critical points. (iii) If −∞ < λk < ∞ for all sufficiently large k and I satisfies (P S), then λk % ∞ as k → ∞. Let us first introduce a special definition useful in stating the main result contained in [11]; see also [10].
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Definition 2.6. Let Q, R, and S be three subsets of X, with R ⊂ Q, and let k be a nonnegative number. We say that (Q, R) links S cohomologically in dimension k over Z2 , if R ∩ S = ∅ and the restriction homomorphism H k (X, X \ S; Z2 ) → H k (Q, R; Z2 ) is not identically zero. In the next statement we summarize Corollary 2.9, Theorem 2.8, and Proposition 2.4 of Degiovanni and Lancelotti [11], and Theorem 6.10 of Degiovanni in [10]. See also Theorem 3.2 and Remark 3.3. of Degiovanni in [10]. This result is crucial in the proof of Theorem 4.3. Theorem 2.7. Let (X, k · kX ) be a Banach space and C− and C+ be two symmetric cones in X such that C+ is closed in X, C− ∩ C+ = {0}, and i(C− \ {0}) = i(X \ C+ ) = k < ∞. Let r− , r+ > 0 and let e ∈ X with −e 6∈ C− . Define the following sets: D− = {u ∈ C− : kukX ≤ r− },
S+ = {u ∈ C+ : kukX = r+ },
Q = {u + te : u ∈ C− , t ≥ 0, ku + tekX ≤ r− }, H = {u + te : u ∈ C− , t ≥ 0, ku + tekX = r− }. Then (Q, D− ∪ H) links S+ cohomologically in dimension k + 1 over Z2 . Let F ∈ C 1 (X) satisfy the (P S) condition. If furthermore sup
F (x) < inf F (x) x∈S+
x∈D− ∪H
and
sup F (x) < ∞, x∈Q
then there exists a critical point u of F at some level c, with inf F (x) ≤ c ≤ sup F (x).
x∈S +
x∈Q
3. The underlying Robin boundary eigenvalue problem Consider the Robin boundary eigenvalue problem associated to (1.1), ( − div(a(x)|∇u|p−2 ∇u) = λf (x)|u|p−2 u in Ω, (3.1) p−2 p−2 a(x)|∇u| ∇u · ν + b(x)|u| u = 0 on ∂Ω, and put λ1 = inf I(u), u∈M
M = {u ∈ E : kukp,f = 1} ,
(3.2)
where I(u) = kukp
and kukp,f =
Z Ω
f (x)|u(x)|p dx
1/p
.
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
We say that λ ∈ R is an eigenvalue for (3.1) and that u ∈ E \ {0} is a corresponding eigenfunction, if Z Z p−2 a(x)|∇u| ∇u · ∇φ dx + b(x)|u|p−2 uφ dS Ω ∂Ω Z (3.3) = λ f (x)|u|p−2 uφ dx Ω
for all φ ∈ E. We call (λ, u) an eigenpair for (3.1). For a wide study on nonlinear eigenvalue problems for the p-Laplace operator via variational methods in bounded domains, under several different boundary conditions of great interest, we refer to [23] and the references therein. Proposition 3.1. The following statements hold. (i) The infimum λ1 in (3.2) is attained; that is, there exists ϕ1 ∈ M which realizes the minimum in (3.2) and represents an eigenfunction for (3.1) corresponding to the eigenvalue λ1 . Moreover, λ1 > 0. (ii) The set E1 of eigenfunctions corresponding to λ1 is a vector space of dimension 1. The elements of E1 are either positive or negative and 1,δ (Ω). of class Cloc (iii) Let v be an eigenfunction of (3.1) corresponding to an eigenvalue λ 6= λ1 . Then v is a nodal solution of (3.1); that is, v changes sign in Ω. Proof. (i) First, I(u) = kukp and J (u) = kukpp,f are continuously Fr´echet differentiable and convex in E. Clearly I0 (0) = J 0 (0) = 0 and J 0 (u) = 0 implies u = 0. In particular, I and J are weak lower semicontinuous on E. Actually, J is weak sequentially continuous on E. Indeed, if (un )n ⊂ E and un * u in E, then un → u in Lp (Ω, f ), being the natural embedding between E and Lp (Ω, f ) compact. Hence J (un ) = kun kpp,f → J (u) = kukpp,f , as claimed. Of course, I is coercive in E, so that an easy contradiction argument shows that I is coercive in E ∩ {u ∈ E : J (u) = kukpp,f ≤ 1}. In conclusion, all the assumptions of Theorem 6.3.2 of [6] are fulfilled, and so λ1 is attained in M. Finally, J (u) = kukpp,f ≤ Bkukp = BI(u) for all u ∈ E, with B = C p and C given in (2.2). Hence, λ1 ≥ 1/B > 0 by (3.2). (ii) Let ϕ1 be an eigenfunction corresponding to λ1 . Since |ϕ1 | is also 1,p a minimizer for I in M, we may assume that ϕ1 ≥ 0. Now ϕ1 ∈ Wloc (Ω) since ϕ1 ∈ E, so that Theorem 2.2 of Pucci and Servadei [35] can be applied with A(x, u, ξ) = −a(x)|ξ|p−2 ξ and B(x, u, ξ) = λf (x)|u|p−2 u, since
Existence theorems for eigenvalue problems
11
the main condition (2.17) of [35] is satisfied because f ∈ L∞ (Ω). Hence ϕ1 ∈ L∞ loc (Ω). From Corollary of Theorem 2 of DiBenedetto [14] it follows 1,δ that ϕ1 ∈ Cloc (Ω), and clearly ϕ1 is a non-negative weak solution of the differential inequality div(a(x)|∇u|p−2 ∇u) ≤ 0 in Ω. Thus ϕ1 > 0 in Ω by the strong maximum principle given in Theorem 5.4.1 of Pucci and Serrin [32] and the Remarks 3 and 4 on page 117 therein, since clearly ϕ1 ∈ C 1 (Ω); cf. also [33]. It remains to show that E1 is a vector space of dimension one. We in some sense follow the main arguments of [2, Theorem 1.1], but for completeness we report here all the principal steps of the proof. Let u and v be two eigenfunctions associated to λ1 . Without loss of generality, we assume that both u and v are positive in Ω. By the above argument, u and v are of class C 1 (Ω) and so the functional u p−1 u p |∇v|p − p |∇v|p−2 ∇v∇u L(u, v) = |∇u|p + (p − 1) v v is well defined and non-negative in Ω as a consequence of Theorem 1.1 of [2]. Since v could be zero on ∂Ω, let us define vn = v + 1/n. Then, by (3.3) with φ = (u/vn )p−1 u ∈ E it follows that Z Z h up i 0≤ L(u, vn ) dx = |∇u|p − |∇v|p−2 ∇ p−1 ∇v dx vn Ω Ω Z Z p 1n u ≤ (3.4) a(x)|∇u|p dx + b(x)v p−1 p−1 dS a0 vn Ω ∂Ω Z o up f (x)v p−1 p−1 dx −λ1 vn Ω Z o o p u 1n 1 n kukp − λ1 f (x)v p−1 p−1 dx → kukp − λ1 kukpp,f = 0 ≤ a0 a0 vn Ω by the Lebesgue dominated convergence theorem and the fact that (λ1 , u) is an eigenpair. Clearly, L(u, vn ) → L(u, v) as n → ∞ in R Ω, since v > 0 in Ω. Therefore by the Fatou lemma and (3.4) we have 0 ≤ Ω L(u, v) dx ≤ 0; that is, L(u, v) = 0 a.e. in Ω. Hence by Theorem 1.1 of [2] there exists k ∈ R such that u = kv in Ω. Of course k > 0, and this shows that E1 has dimension 1. (iii) Suppose for the sake of contradiction that v does not change sign in Ω, and let u be an eigenfunction of (3.1) corresponding to λ1 . Without loss of generality we assume that both u and v are positive in Ω. Using the first argument of step (ii) we get that u and v are of class C 1 (Ω) and so, proceeding as above, we find again (3.4). Hence L(u, v) = 0 a.e. in Ω; that
12
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
is, there exists k > 0 such that u = kv in Ω. In particular, v ∈ E1 . This is impossible, because λ 6= λ1 . Therefore, v is a nodal solution of (3.1). The next step is to prove that the spectrum of the Robin problem (3.1) is closed. For similar results, when Ω is bounded and E = W 1,p (Ω), we refer to Lemma 2.3 of [23]. To this aim, let us first establish a preliminary result. Lemma 3.2. The C 1 functional I has the property that for all u, v ∈ E hI0 (u) − I0 (v), u − vi ≥ p (kukp−1 − kvkp−1 ) (kuk − kvk) ≥ 0. Proof. Straightforward computations yield hI0 (u) − I0 (v), u − vi Z = p a(x) |∇u|p + |∇v|p − |∇u|p−2 ∇u · ∇v − |∇v|p−2 ∇v · ∇u dx Ω Z +p b(x) |u|p + |v|p − |u|p−2 uv − |v|p−2 vu dS ∂Ω Z ≥ p a(x) |∇u|p + |∇v|p − |∇u|p−1 |∇v| − |∇v|p−1 |∇u| dx Ω Z +p b(x) |u|p + |v|p − |u|p−1 |v| − |v|p−1 |u| dS. ∂Ω
By H¨ older’s inequality we obtain Z Z 1/p0 Z 1/p p−1 p a(x)|∇u| |∇v| dx ≤ a(x)|∇u| dx a(x)|∇v|p dx Ω
Ω
Ω
and Z
p−1
b(x)|u|
|v| dS ≤
∂Ω
Z
p
b(x)|u| dS
1/p0 Z
∂Ω
b(x)|v|p dS
1/p
.
∂Ω
Applying the inequality aα c1−α + bα d1−α ≤ (a + b)α (c + d)1−α , which holds for any α ∈ (0, 1) and for any a, b, c, d > 0, with α = 1/p0 , Z Z a= a(x)|∇u|p dx, b = b(x)|u|p dS, Ω ∂Ω Z Z c= a(x)|∇v|p dx, d = b(x)|v|p dS, Ω
∂Ω
we conclude that Z Z a(x)|∇u|p−1 |∇v| dx + Ω
∂Ω
b(x)|u|p−1 |v| dS ≤ kukp−1 kvk.
Existence theorems for eigenvalue problems
Similarly, we have Z Z p−1 a(x)|∇v| |∇u| dx +
13
b(x)|v|p−1 |u| dS ≤ kvkp−1 kuk.
∂Ω
Ω
This concludes the proof.
Proposition 3.3. The following statements hold. (i) The set of the eigenvalues of (3.1) is closed in E. (ii) λ1 is isolated; i.e., there exists δ > 0 such that the open interval (0, λ1 + δ) does not contain any eigenvalue of (3.1) other than λ1 . Proof. We somehow follow the main arguments of [23, Theorems 5.9, 5.13] and [20, Proposition 3.1], based on the techniques developed by Lindqvist in [24]. For completeness we report here all the principal steps of the proof. (i) Let n 7→ (µn , un ) be a sequence of eigenpairs of (3.1), with µn → λ ≥ 0 as n → ∞. Without loss of generality we assume that kun k = 1 for all n and that (un )n converges weakly to some u ∈ E. By (3.3) with φ = un − u, we obtain hI0 (un ) − I0 (u), un − ui = pµn hF(un ), un − ui − hI0 (u), un − ui → 0 as n → ∞ by Proposition 2.3. Hence kun k → kuk as n → ∞ by Lemma 3.2. In conclusion, un → u in E by the Clarkson uniform convexity Proposition A.2 of E. Therefore, hI0 (un ), φi → hI0 (u), φi and hF(un ), φi → hF(u), φi for all φ ∈ E, by the fact that I ∈ C 1 (E) and by Proposition 2.3. Since for all n and all φ ∈ E hI0 (un ), φi = pµn hF(un ), φi by (3.3), passing to the limit as n → ∞, we get the claim at once. (ii) Assume for the sake of contradiction that λ1 is not isolated. Then by (i) there exists a sequence of eigenpairs n 7→ (µn , un ) such that un → u, µn → λ1 as n → ∞ and (λ1 , u) is an eigenpair. Without loss of generality we assume that kun k = kuk = 1 and that u > 0 in Ω by Proposition 3.1-(ii). Hence u = ϕ1 and there exists δ > 0 such that µn ∈ (λ1 , λ1 + δ) for all n. Put Un− = {x ∈ Ω : un (x) < 0} for all n. Clearly meas(Un− ) > 0 by Proposition 3.1-(iii). Since (µn , un ) is an eigenpair for (3.1), by (3.3), with φ = u− n = min{un , 0} ∈ E, it follows that Z Z Z p − p − p p ku− k = a(x)|∇u | dx + b(x)|u | dS = µ f (x)|u− n n n n n | dx, Un−
∂Ω∩Un−
Un−
which gives by (2.1) and the fact that µn ∈ (λ1 , λ1 + δ), p − p ku− n k ≤ ckun kLp (U − ,w ) , n
1
14
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
where c = Cf (λ1 + δ) > 0. Since meas (Un− ) > 0, then ku− n k > 0. Thus, p − p taking ε = 1/2c and Rn > 0 so large that kun kLp (U − ,w ) − εku− nk < n
1
p N ku− n kLp (Un− ∩B ,w1 ) , where BRn is the ball of center 0 and radius Rn of R . Rn Now, combining the above inequalities, we get Z Z p/N 1 − p − p − p kun k < c w1 (x)|un | dx ≤ ckun kp∗ ,w1 (1 + |x|)−α1 dx 2 Un− ∩BRn Un− ∩BRn p/N p − ≤ dku− , n k meas(Un ∩ BRn )
where d = cw1 c > 0 and cw1 > 0 is such that kvkpp∗ ,w1 ≤ cw1 kvkp for all v ∈ E. − In other words, recalling that ku− n k > 0, we have that inf n meas(Un ∩BRn ) ≥ −N/p σ > 0, where σ = (2d) . In particular, there exists R > 0 such that inf n meas(BR ∩ Un− ∩ BRn ) ≥ σ/2. Since kun − ϕ1 k → 0 as n → ∞, then kun − ϕ1 kp∗ ,w1 → 0 as n → ∞. Thus, by Lemma 2.1 and Egorov’s theorem, up to a subsequence, un → ϕ1 quasi-uniformly in Ω∩BR . This is impossible, since ϕ1 > 0 by (ii). Indeed, since ϕ1 is continuous and positive in Ω, there exists ε > 0 such that meas(Ωε ) > meas(Ω ∩ BR ) − σ/4, where Ωε = {x ∈ Ω ∩ BR : ϕ1 (x) > ε}. Moreover, since un → ϕ1 quasiuniformly in Ω ∩ BR , there exist Aε ⊂ Ωε and nε such that meas(Aε ) > meas(Ωε )−σ/4 and |un (x)−ϕ1 (x)| ≤ ε/2 for all x ∈ Aε and all n ≥ nε . Thus, unε ≥ ϕ1 − ε/2 > ε/2 in Aε ⊂ Un+ε ∩ BR , where Un+ε = {x ∈ Ω : unε (x) > 0}. Hence, since BR ∩ Un−ε ∩ Bnε ⊂ (Ω ∩ BR ∩ Bnε ) \ (Un+ε ∩ BR ) ⊂ (Ω ∩ BR ) \ Aε , we get meas(BR ∩ Un−ε ∩ BRnε ) ≤ meas(Ω ∩ BR ) − meas(Aε ) < meas(Ω ∩ BR ) − meas(Ωε ) + σ/4 < meas(Ω ∩ BR ) − meas(Ω ∩ BR ) + σ/2 = σ/2. This contradiction shows that λ1 is isolated.
The set M defined in (3.2) is a closed Z2 -invariant Finsler manifold of E of class C 1 . Of course M = 6 ∅, the space E being compactly embedded in Lp (Ω, f ). Furthermore, I(u) = kukp is even and of class C 1 (E) and so of class C 1 (M). Let Fk and λk be respectively the corresponding sets and numbers defined as in Proposition 2.5. Now, by Proposition 3.1 of [11] we get that i(A) = sup{i(K) : K ⊂ A is compact and symmetric}
(3.5)
Existence theorems for eigenvalue problems
15
for all open symmetric subsets A of M. Hence λk = inf max I(u), K∈Gk u∈K
Gk = {K ⊂ M : K is compact and symmetric, with i(K) ≥ k}.
(3.6)
To prove the validity of (3.6) we first show that Gk = 6 ∅ for all k. Fix k ∈ N. Since meas({x ∈ Ω : f (x) > 0}) > 0, there exist k open balls B1 , . . . , Bk such that Bi ∩ Bj = ∅ if i 6= j and meas({x ∈ Ω : f (x) > 0} ∩ Bi ) > 0
for all i = 1, . . . , k.
Choose ui ∈ C0∞ (RN ), with supp ui ⊂ Ω ∩ Bi and J (ui ) > 0 for every i = 1, . . . , k. Normalize ui so that J (ui ) = 1, and let Ek be the span of P {ui }ki=1 . For every u ∈ Ek , we have u = ki=1 ai ui and Z J (u) = ∪ki=1
Z k k X p X p f (x) ai ui dx = |ai |
supp ui
i=1
i=1
f (x)|ui |p dx =
supp ui
k X
|ai |p .
i=1
1/p defines a norm on E which is equivalent to the norm Hence, k u 7→ J (u) k · k E , Ek being finite dimensional. Therefore, K = {u ∈ Ek : J (u) = 1} is k compact and symmetric, and i(K) = k by (i4 ). In conclusion, K ∈ Gk 6= ∅, as required. Now, call the right-hand side of (3.6) µk . Of course, µk ≥ λk . If µk > λk , there would exist M ∈ Fk such that M ⊂ Ik = {u ∈ M : I(u) < µk } and i(M ) ≥ k. Hence, there exists a compact and symmetric subset K of Ik , with i(K) ≥ k by (3.5). Therefore, we get the required contradiction µk < µk , and so (3.6) holds. In particular, λ1 > 0 coincides with the number defined in (3.2). Indeed, ˜ 1 denotes the number in (3.6) when k = 1, we get immediately λ ˜1 = if λ inf K∈G1 I(uK ) ≤ λ1 , since M ∈ G1 . On the other hand, I(uK ) ≥ I(ϕ1 ) = λ1 for all K ∈ G1 , since K ⊂ M and ϕ1 is the first eigenfunction in Proposi˜ 1 ≥ λ1 and in turn λ ˜ 1 = λ1 , as claimed. tion 3.1-(i). Therefore, λ p −1 Since M = J (1) and J (u) = kukp,f , u ∈ M is a critical point of I M if and only if there is some µ ∈ R such that I0 (u) = µJ 0 (u) and J 0 (u) = pF(u). Clearly, I0 (u) 6= 0 whenever u ∈ M, so that µ 6= 0. Actually I0 (u) = 0 if and + only if u = 0 ∈ E. Similarly, every λk ∈ R for all k. Now, if we show that I M satisfies the (P S)c condition for all c ∈ R+ , we are in a position to apply the crucial Proposition 2.5. Following the main ideas of [11] we prove:
16
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
Proposition 3.4. I M satisfies the (P S)c condition for all c, λ1 = min I(u), u∈M
λk ∈ R+ ,
lim λk = ∞.
k→∞
For all k, with λk < λk+1 , we have i ({u ∈ E \ {0} : I(u) ≤ λk J (u)}) = i ({u ∈ E : I(u) < λk+1 J (u)}) = k. Proof. Let c ∈ R and (un )n ⊂ M be a (P S)c sequence of I M . Hence I(un ) → c and I0 (un ) − pµn F(un ) → 0 in E ? for a suitable sequence (µn )n in R. Clearly, (un )n is bounded in E, and so passing to a subsequence un * u in E, the space E being reflexive. Hence, I(un ) → I(u) in L1 (Ω) by Proposition 2.3 with w = f , so that kukp,f = 1, and in turn u 6= 0 and I(u) > 0. Now, p (I(un ) − µn ) = hI0 (un ), un i − pµn hF(un ), un i → 0. Hence, µn → c and 0 < I(u) ≤ c by the weak lower semicontinuity of the norm in E. Therefore, c > 0. Taking φ = un − u ∈ E in (3.3), we obtain hI0 (un ), un − ui − pµn hF(un ), un − ui = hI0 (un ) − I0 (u), un − ui − pµn hF(un ) − F(u), un − ui + hI0 (u), un − ui − pµn hF(u), un − ui for all n ∈ N. Consequently, 0 ≤ hI0 (un ) − I0 (u), un − ui → 0 as n → ∞ by Proposition 2.3. Clearly, hI0 (u), un − ui → 0 as n → ∞, and un * u in E. Hence hI0 (un ), un − ui → 0 as n → ∞. (3.7) Moreover, I(u) ≤ lim inf n I(un ) due to the weak lower semicontinuity of I on E. On other hand, by convexity, I(u) + hI0 (un ), un − ui ≥ I(un ), so that I(u) ≥ lim supn I(un ). In other words, I(u) = limn I(un ). Thanks to the Clarkson uniform convexity Proposition A.2 of E, it follows that kun − uk → 0 as n → ∞, as required. In other words, I M satisfies the (P S)c condition for all c ∈ R+ . An application of Proposition 2.5 shows that λk → ∞ as k → ∞. Assume now that k ≥ 1 and λk < λk+1 . Put C = {u ∈ M : I(u) ≤ λk }
and U = {u ∈ M : I(u) < λk+1 }.
From now on the proof can follow word by word the proof of Theorem 3.2 of [11] by virtue of Proposition 2.3.
Existence theorems for eigenvalue problems
17
4. Existence in the Ambrosetti–Rabinowitz case Throughout the section we assume that →R R sg : Ω×R → R and h : ∂Ω×R Rs are Carath´eodory functions, G(x, s) = 0 g(x, t) dt, H(x, s) = 0 h(x, t)dt, g0 , g1 , γ0 , and γ1 are measurable functions on Ω, while h0 , h1 , τ0 , and τ1 are measurable functions on ∂Ω, such that (g1 ) |g(x, s)| ≤ g0 (x)|s|p−1 + g1 (x)|s|r−1 for a.a. x ∈ Ω and all s ∈ R, where p < r < p∗ = pN/(N − p), with 0 < g0 (x) ≤ Cg w2 (x)
and
0 ≤ g1 (x) ≤ Cg g0 (x) for a.a. x ∈ Ω,
g0 /w2 ∈ Lr/(r−p) (Ω, w2 ),
g0 ∈ Lr˜/(˜r−p) (Ω, w2 ),
where w2 (x) = (1 + |x|)−α2 , N − r(N − p)/p < α2 < N , and r˜ is an appropriate exponent, with r < r˜ < min{pr, p∗ }; G(x, s) G(x, s) = 0 and lim = ∞, both uniformly in Ω \ N , (g2 ) lim p s→0 g0 (x)|s| |s|→∞ g0 (x)|s|p with meas(N ) = 0; (g3 ) γ0 ∈ L1 (Ω) and 0 < γ1 (x) ≤ Cγ w1 (x) a.e. in Ω, and there exists µ > p such that for a.a. x ∈ Ω and all s ∈ R µG(x, s) ≤ sg(x, s) + γ0 (x) + γ1 (x)|s|p ; (g4 ) G(x, s) ≥ 0 for a.a. x ∈ Ω and all s ∈ R; (h1 ) |h(x, s)| ≤ h0 (x)|s|p−1 + h1 (x)|s|q−1 for a.a. x ∈ ∂Ω and all s ∈ R, where p < q < p∗ = p(N − 1)/(N − p), with 0 < h0 (x) ≤ Ch w3 (x)
and
q/(q−p)
h0 /w3 ∈ L
0 ≤ h1 (x) ≤ Ch h0 (x) for a.a. x ∈ ∂Ω,
(∂Ω, w3 ),
h0 ∈ Lq˜/(˜q−p) (∂Ω, w3 ),
where w3 (x) = (1 + |x|)−α3 , N − 1 − q(N − p)/p < α3 < N , and q˜ is an appropriate exponent, with q < q˜ < min{qr, p∗ }; H(x, s) H(x, s) = 0 and lim = ∞, both uniformly in ∂Ω\N, (h2 ) lim s→0 h0 (x)|s|p |s|→∞ h0 (x)|s|p with measN −1 (N) = 0; (h3 ) w4 (x) = (1 + |x|)−α4 , p − 1 < α4 < N , τ0 ∈ L1 (∂Ω), 0 < τ1 (x) ≤ Cτ w4 (x) a.e. in ∂Ω, and there exists µ ˜ > p such that µ ˜H(x, s) ≤ sh(x, s)+τ0 (x)+τ1 (x)|s|p for a.a. x ∈ ∂Ω and all s ∈ R; (h4 ) H(x, s) ≥ 0 for a.a. x ∈ ∂Ω and all s ∈ R. 0
1/(1−r)
Note that if g0 ∈ Lr (Ω, w2 ), then g0 /w2 ∈ Lr/(r−p) (Ω, w2 ) and g0 ∈ 0 1/(1−q) Lr˜/(˜r−p) (Ω, w2 ). Also h0 ∈ Lq (∂Ω, w3 ) yields h0 /w3 ∈ Lq/(q−p) (∂Ω, w3 )
18
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
and h0 ∈ Lq˜/(˜q−p) (∂Ω, w3 ). Therefore, conditions (g1 ) and (h1 ) are weaker than the corresponding assumption A1 in [31]; see also [27]. By (g1 ) and (h1 ) we have g(x, 0) = 0 for a.a. x ∈ Ω and h(x, 0) = 0 for a.a. x ∈ ∂Ω, so that problem (1.1) always admits the trivial solution u ≡ 0. Let Φ : E → R be the Euler–Lagrange functional associated to (1.1) defined for all u ∈ E by Z Z 1 λ Φ(u) = kukp − kukpp,f − G(x, u) dx − H(x, u) dS. (4.1) p p Ω ∂Ω The embeddings given in Section 2 assure that Φ is well-defined and of class C 1 (E). A weak solution of (1.1) is a function u ∈ E such that for all φ ∈ E Z Z a(x)|∇u|p−2 ∇u · ∇φ dx + b(x)|u|p−2 uφ dx Ω Z Z∂Ω Z p−2 = λ f (x)|u| uφ dx + g(x, u)φ dx + h(x, u)φ dS. Ω
Lemma 4.1. (i)
Ω
kuk−p
Z
∂Ω
Z G(x, u) dx +
Ω
H(x, u) dS → 0 as kuk → 0.
∂Ω
(ii) Let w and w ˜ be two weights on Ω and on ∂Ω, respectively, such that the embeddings E ,→ Lp (Ω, w) and E ,→ Lp (∂Ω, w) ˜ are compact. Let (un )n be a sequence in E such that kun k ≤ c (kun kp,w + kun kp,w,∂Ω ) for all n and ˜ some constant c > 0 independent of n. If kun k → ∞, then Z Z −p kun k G(x, un ) dx + H(x, un ) dS → ∞ as n → ∞. Ω
∂Ω
Proof. (i) By (g1 ) and (g2 ) clearly G0 (x, s), defined by G0 (x, s) = G(x, s)/w2 (x)|s|p if s 6= 0 and G0 (x, 0) = 0 if s = 0, is a Carath´eodory function in Ω × R. Moreover, using also (g4 ), for a.a. x ∈ Ω and all s ∈ R we have by (g1 ) 0 ≤ G0 (x, s) ≤ C0 g˜0 (x) + |s|r−p , where C0 = max{1/p, Cg2 /r} and g˜0 = g0 /w2 . Therefore, applying the H¨ older inequality, we obtain Z Z (r−p)/r 0≤ G(x, u) dx ≤ w2 (x)G0 (x, u)r/(r−p) dx kukpr,w2 , Ω
with
Ω
Z Ω
w2 (x)G0 (x, u)r/(r−p) dx → 0
as kuk → 0.
Existence theorems for eigenvalue problems
19
Indeed, by Lemma 2.2-(i), with A = Ω, d = N , w = w2 , α = r, and β = r/(r − p), the Nemytskii operator NG0 (u) = G0 (x, u), NG0 (0) = 0, maps continuously Lr (Ω, w2 ) into Lr/(r−p) (Ω, w2 ). Therefore, Z G(x, u) dx = o(kukp ) as kuk → 0, Ω r L (Ω, w2 ) even
since E ,→ compact by Theorem 1 of [31]. Similarly, by (h1 ) and (h2 ), clearly, H0 (x, s), defined by H0 (x, s) = H(x, s)/w3 (x)|s|p if s 6= 0 and H0 (x, 0) = 0 if s = 0, is a Carath´eodory function in ∂Ω × R. Moreover, using also (h4 ), for a.a. x ∈ ∂Ω and all s ∈ R, we obtain by (h1 ) ˜ 0 (x) + |s|q−p , 0 ≤ H0 (x, s) ≤ C˜0 h ˜ 0 = h0 /w3 . Therefore, applying the where C˜0 = max{1/p, Ch2 /q} and h H¨ older inequality, we obtain Z Z (q−p)/q 0≤ H(x, u) dS ≤ w3 (x)H0 (x, u)q/(q−p) dS kukpq,w3 ,∂Ω , ∂Ω
∂Ω
with Z
w3 (x)H0 (x, u)q/(q−p) dS → 0
as kuk → 0.
∂Ω
Indeed, by Lemma 2.2-(i), with A = ∂Ω, d = N − 1, w = w3 , α = q, and β = q/(q −p), the Nemytskii operator NH0 (u) = H0 (x, u), NH0 (0) = 0, maps continuously Lq (∂Ω, w3 ) into Lq/(q−p) (∂Ω, w3 ), using the argument shown above. Therefore, Z H(x, u) dS = o(kukp ) as kuk → 0, ∂Ω
Lq (∂Ω, w3 )
since E ,→ even compact by Theorem 1 of [31]. In conclusion, the assertion follows at once. (ii) Let (un )n be a sequence in E as in the statement, so that vn = un /kun k is in the unit sphere of E for all n sufficiently large. Therefore, up to a subsequence, still denoted for simplicity by (vn )n , there is v ∈ E such that vn * v in E, vn → v in Lp (Ω, w), vn → v in Lp (∂Ω, w), ˜ vn → v a.e. in Ω, and vn → v a.e. in ∂Ω. From the second and third parts of Proposition 2.3 we get 1 ≤ c lim (kvn kp,w + kvn kp,w,∂Ω ) = c (kvkp,w + kvkp,w,∂Ω ). ˜ ˜ n
20
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
Hence either |v| > 0 in a subset A of Ω, with meas(A) > 0, or |v| > 0 in a subset Γ of ∂Ω, with measN −1 (Γ) > 0, since c > 0 and w > 0 a.e. in Ω and w ˜ > 0 a.e. in ∂Ω. Case |v| > 0 in A, meas(A) > 0. For n sufficiently large G(x, un (x)) G(x, kun kvn (x)) = g0 (x)|vn (x)|p kun kp g0 (x)kun kp |vn (x)|p
a.e. in A.
Hence limn kun k−p G(x, un (x)) = ∞ a.e. in A by (g2 ). By (g4 ) and the Fatou lemma we get Z −p G(x, un ) dx → ∞ as n → ∞. kun k Ω
Case |v| > 0 in Γ, measN −1 (Γ) > 0. Similarly, for n sufficiently large H(x, un (x)) H(x, kun kvn (x)) = h0 (x)|vn (x)|p kun kp h0 (x)kun kp |vn (x)|p
a.e. in Γ.
Thus limn kun k−p H(x, un (x)) = ∞ a.e. in Γ by (h2 ). By (h4 ) and the Fatou lemma we have Z −p kun k H(x, un ) dS → ∞ as n → ∞. ∂Ω
From (g4 ) and (h4 ) we get the assertion at once.
Lemma 4.2. (i) If (un )n is bounded in Lr˜(Ω, w2 ) and converges to some 0 u ∈ Lr˜(Ω, w2 ) a.e. in Ω, then Ng (un ) → Ng (u) in Lr˜ (Ω, w2 ), and so in E ? . (ii) If (un )n is bounded in Lq˜(∂Ω, w3 ) and converges to some u ∈ Lq˜(∂Ω,w3 ) 0 a.e. in ∂Ω, then Nh (un ) → Nh (u) in Lq˜ (∂Ω, w3 ), and so in E ? . (iii) For all λ and c in R the functional Φ satisfies the (P S)c condition. Proof. (i) From (g1 ) we clearly get |s|1−˜r g(x, s) → 0 as |s| → ∞ uniformly in Ω \ N for some N , with meas(N ) = 0. Hence, for all ε > 0 there is Cε > 0 such that |g(x, s)| ≤ Cε g0 (x)|s|p−1 + ε|s|r˜−1
a.e. in Ω and for all s ∈ R.
Applying Young’s inequality, we get r˜ − p Cε g0 (x) (˜r−1)/(˜r−p) p − 1 (˜r−1)/(p−1) + ε + ε |s|r˜−1 . |g(x, s)| ≤ r˜ − 1 ε r˜ − 1 Hence, by Lemma 2.2-(ii), with A = Ω, d = N , w = w2 , α = r˜, and β = r˜0 , we get the assertion at once, since g0 ∈ Lr˜/(˜r−p) (Ω, w2 ) by (g1 ). 0 Naturally the strong convergence in Lr˜ (Ω, w2 ) implies convergence in E ? , since E ,→ Lr˜(Ω, w2 ) compact.
Existence theorems for eigenvalue problems
21
(ii) Similarly, |s|1−˜q h(x, s) → 0 as |s| → ∞ uniformly in Ω \ N for some N, with meas(N) = 0, by (h1 ). Thus, for all ε > 0 there is Kε > 0 such that |h(x, s)| ≤ Kε h0 (x)|s|p−1 + ε|s|q˜−1
a.e. in ∂Ω and for all s ∈ R.
Proceeding as above and applying Young’s inequality we obtain q˜ − p Kε h0 (x) (˜q−1)/(˜q−p) p − 1 (˜q−1)/(p−1) |h(x, s)| ≤ ε + ε |s|q˜−1 . + q˜ − 1 ε q˜ − 1 Hence, by Lemma 2.2-(ii), with A = ∂Ω, d = N − 1, w = w3 , α = q˜, and β = q˜0 , we reach the conclusion, since h0 ∈ Lq˜/(˜q−p) (∂Ω, w3 ) by (h1 ). 0 Naturally, the strong convergence in Lq˜ (∂Ω, w3 ) implies convergence in E ? , since E ,→ Lq˜(∂Ω, w3 ) compact. (iii) Let (un )n ⊂ E be a (P S)c sequence for Φ. We claim that (un )n is bounded in E. Assume for the sake of contradiction that kun k → ∞ as n → ∞ up to a subsequence. Case µ ≤ µ ˜. By (g3 ), (h3 ), and (h4 ), we have µ − 1 kun kp − λkun kpp,f µΦ(un ) − hΦ0 (un ), un i ≥ p Z Z [µG(x, un ) − un g(x, un )] dx − [˜ µH(x, un ) − un h(x, un )] dS − ∂Ω Ω Z ≥ 2κkun kp − [γ0 (x) + {2|λ|κf (x) + γ1 (x)}|un |p ] dx Ω Z − [τ0 (x) + τ1 (x)|un |p ] dS, ∂Ω
where 2κ = µ/p − 1 > 0. Now, kun k → ∞ as n → ∞, so that for all n sufficiently large Z Z 0 γ0 (x) dx + τ0 (x) dS ≤ κkun kp . µΦ(un ) − hΦ (un ), un i + Ω
∂Ω
Combining together the last two inequalities, we get κkun kp ≤ kun kpp,w + kun kpp,w,∂Ω , ˜
w = 2|λ|κf + γ1 ,
w ˜ = τ1 ,
(4.2)
for all n large enough. Case µ > µ ˜. Similarly, by (g3 ), (h3 ), and (g4 ), we get µ ˜ µ ˜Φ(un ) − hΦ0 (un ), un i ≥ − 1 kun kp − λkun kpp,f p Z Z − [µG(x, un ) − un g(x, un )] dx − [˜ µH(x, un ) − un h(x, un )] dS Ω
∂Ω
22
Giuseppina Autuori, Patrizia Pucci and Csaba Varga p
Z
[γ0 (x) + {2|λ|κf (x) + γ1 (x)}|un |p ] dx Ω Z [τ0 (x) + τ1 (x)|un |p ] dS, −
≥ 2κkun k −
∂Ω
where now 2κ = µ ˜/p − 1 > 0. Since kun k → ∞ as n → ∞, for all n sufficiently large Z Z µ ˜Φ(un ) − hΦ0 (un ), un i + γ0 (x) dx + τ0 (x) dS ≤ κkun kp . Ω
∂Ω
Combining together the last two inequalities, we again obtain (4.2). In both cases the embeddings E ,→ Lp (Ω, w) and E ,→ Lp (∂Ω, w) ˜ are compact by (2.1), (g3 ), and (h3 ). Hence, applying Lemma 4.1-(ii), with w = 2|λ|κf + γ1 , w ˜ = τ1 , and c = 1/κ > 0, we get Z Z kun k−p G(x, un ) dx + H(x, un ) dS → ∞ as n → ∞. Ω
∂Ω
In conclusion, since kun kp,f ≤ Ckun k for all n by (2.2), we have Φ(un ) kun kp R R kun kp 1 p,f Ω G(x, un ) dx + ∂Ω H(x, un ) dS ≤ + lim |λ| − = −∞, p n→∞ kun kp kun kp
0 = lim
n→∞
which is the required contradiction. Therefore, (un )n is bounded in E, and so there exists u ∈ E such that, up to a subsequence, un * u in E, un → u in Lr˜(Ω, w2 ), un → u in Lq˜(∂Ω, w3 ), un → u a.e. in Ω, and un → u a.e. in ∂Ω. Now, 1 0 hI (un ), un − ui − λhF(un ), un − ui p Z Z − g(x, un )(un − u) dx − h(x, un )(un − u) dS → 0,
hΦ0 (un ), un − ui =
Ω
∂Ω
since (un )n is a (P S)c sequence. Hence, by virtue of (i) and (ii) of this lemma and of Proposition 2.3, we have shown that hI0 (un ), un − ui → 0
as n → ∞;
that is, (3.7) holds. This implies that actually un → u in E as n → ∞, using the same argument given in the proof of Proposition 3.4. The lemma is now proved.
Existence theorems for eigenvalue problems
23
Theorem 4.3. Under assumptions (g1 )–(g4 ) and (h1 )–(h4 ) problem (1.1) admits at least a nontrivial solution u ∈ E for every λ ∈ R. Proof. Consider first the case λ ≥ λ1 . By Propositions 3.1, 3.3, and 3.4 we have λ1 < λ2 ≤ · · · ≤ λk · · · → ∞, so that there exists k ≥ 1 such that λk ≤ λ < λk+1 . Define the two symmetric closed cones C− = {u ∈ E : I(u) ≤ λk J (u)} ,
C+ = {u ∈ E : I(u) ≥ λk+1 J (u)} .
Clearly, C− ∩ C+ = {0} and i(C− \ {0}) = i(E \ C+ ) = k by Proposition 3.4. Furthermore, for all u ∈ C+ Z Z 1 λ Φ(u) ≥ 1− kukp − G(x, u) dx − H(x, u) dS. (4.3) p λk+1 Ω ∂Ω Thus, by Lemma 4.1-(i) there exists a number r+ > 0 such that Φ(u) ≥ α p for all u ∈ C+ , with kuk = r+ , where α = r+ (1 − λ/λk+1 )/2p > 0. On the p other hand, Φ(u) ≤ (λk − λ)kukp,f /p ≤ 0 for all u ∈ C− by (g4 ) and (h4 ). Since E ,→ Lp (Ω, f ) and E ,→ Lp (∂Ω, w4 ) are compact by Theorem 1 of [31], the cone C− is also closed in the real normed space E = (E, |||·|||), where |||·||| = k · kp,f + k · kp,w4 ,∂Ω . Taking e ∈ E \ C− and t > 0, we easily see as in [11] that 1/p
ku + tek ≤ t max{λk , kek/ |||e|||}(|||u/t||| + |||e|||). Moreover, by Proposition 2.12. of [11] there exists β ≥ 1 such that for all u ∈ C− we have |||u/t||| + |||e||| ≤ β |||u/t + e|||. In conclusion, ku + tek ≤ κ |||u + te|||
(4.4)
1/p
for all u ∈ C− and t ≥ 0, where κ = β max{λk , kek/ |||e|||} > 0. Now, along any sequence (un )n ⊂ C− + R+ e ⊂ E such that kun k → ∞ Lemma 4.1-(ii) can be applied, since kun k ≤ κ (kun kp,f + kun kp,w4 ,∂Ω ) for all n. Therefore, an easy contradiction argument shows the existence of some r− > r+ such that Φ(u) ≤ 0 for all u ∈ C− + R+ e, with kuk ≥ r− . The geometrical construction of Theorem 2.7 is completed, so that the corresponding sets satisfy the assertion. In particular, (Q, D− ∪ H) links S+ cohomologically in dimension k + 1 over Z2 and sup u∈D− ∪H
Φ(u) ≤ 0 < α ≤ inf Φ(u) u∈S+
and
p sup Φ(u) ≤ r− /p < ∞.
(4.5)
u∈Q
Finally, Φ ∈ C 1 (E) satisfies the (P S)c condition by Lemma 4.2-(iii) for all λ and c in R. Therefore, problem (1.1) admits a weak nontrivial solution u ∈ E, with Φ(u) ≥ α by virtue of Theorem 2.7.
24
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
On the other hand, if λ < λ1 , the geometrical structure of the mountainpass theorem of Ambrosetti and Rabinowitz in [3] is valid. Indeed, Φ(0) = 0 and for all u ∈ E Z Z 1 λ p Φ(u) ≥ kuk − G(x, u(x)) dx − H(x, u) dS, 1− p λ1 Ω ∂Ω so that by virtue of Lemma 4.1-(i) there exists r > 0 sufficiently small so that Φ(u) ≥ αr for all u ∈ E, with kuk = r, where αr = rp (1−λ/λ1 )/2p > 0. Fix a finite-dimensional subspace W of E and w ∈ W , with kwk = 1. Put n 7→ un = nw ∈ W . Then kun k ≤ cW (kun kp,f + kun kp,w4 ,∂Ω ) for all n, where cW > 0 is independent of n. This can be done, W being a finitedimensional subspace of E. Since kun k → ∞ as n → ∞, by Lemma 4.1-(ii) and by (2.2) R R p − p max{1, C λ } Ω G(x, un (x)) dx + ∂Ω H(x, u) dS − → −∞ Φ(un ) ≤ kun k p kun kp as n → ∞. Hence, fix n so large that v = nw ∈ W is such that kvk > r and Φ(v) < 0. Finally, define c = inf
max Φ(u),
γ∈Γ u∈γ([0,1])
where Γ = {γ ∈ C([0, 1]) : γ(0) = 0, γ(1) = v}.
Then c ≥ αr and c is a critical value for Φ in E by virtue of Theorem 2.1 of [3]. In other words, problem (1.1) admits a weak nontrivial solution u ∈ E, with Φ(u) ≥ αr . 5. The Nehari manifold method In the second part of the paper we plan to face (1.1), when λ < λ1 , with the Nehari manifold method, developed by Szulkin and Weth in [38]. In this section we describe it in an abstract setting. Let E be a real uniformly convex Banach space and SE be its unit sphere. Let Φ ∈ C 1 (E), and suppose that u 6= 0 is a critical point of Φ; i.e., Φ0 (u) = 0. Then necessarily u is contained in the set N = {u ∈ E \{0} : hΦ0 (u), ui = 0}. Therefore, N is a natural constraint for the problem of finding nontrivial critical points of Φ. The set N is called a Nehari manifold, even if in general it may not be a manifold. Set c = inf u∈N Φ(u). Clearly, if c is attained at some u0 ∈ N , then u0 is a nontrivial critical point of Φ in E. Since u0 is a solution of the equation Φ0 (u) = 0, with “minimal energy” Φ, we call u0 a ground state of Φ. Before proving the main theorems of the section, let us start with the structural assumptions (i)–(vi) below and some preliminary useful results.
Existence theorems for eigenvalue problems
25
Let I, I0 , I0 , and J0 be functionals of class C 1 (E), satisfying (i) kI 0 (u)kE ? = o(kukp−1 ) as kuk → 0; (ii) s 7→ s1−p hI 0 (su), ui is strictly increasing for u 6= 0 and s > 0; (iii) s−p I(su) → ∞ as s → ∞, uniformly for u on weakly compact subsets of E \ {0}; (iv) I 0 is completely continuous; (v) I0 is positively homogeneous of degree p > 1 and satisfies c0 kukp ≤ I0 (u) ≤ C0 kukp
(5.1)
for some 0 < c0 ≤ C0 and for all u ∈ E; (vi) I0 = I0 + J0 , where J00 is completely continuous, while I0 is weakly lower semicontinuous and I00 satisfies hI00 (u) − I00 (v), u − vi ≥ c1 (kukp−1 − kvkp−1 )(kuk − kvk)
(5.2)
for some c1 > 0 and for all u, v ∈ E. Put Φ = I0 − I. Hence, Φ(0) = 0, since I0 (0) = 0 by (v) and I(0) = 0 by (i). The next lemma is proved following some of the arguments given in [39, Lemma 4.1]. Lemma 5.1. The following properties hold. (a) For all v ∈ E \ {0} there exists a unique sv > 0 such that sv v ∈ N and Φ(sv v) = max Φ(sv) > 0; s>0
(b) the function η : E \ {0} → R+ , η(v) = sv , where sv is uniquely defined by (a), is continuous; (c) the map m : SE → N , m(z) = sz z, where sz is uniquely defined by (a), is a homeomorphism; (d) there exists δ > 0 such that sz ≥ δ for all z ∈ SE and inf kuk = u∈N
inf ksz zk > 0;
z∈SE
(e) c = inf Φ(u) = u∈N
inf
max Φ(sv) = inf max Φ(sz);
v∈E\{0} s>0
z∈SE s>0
(f ) if c is attained, then c > 0. Proof. (a) Fix v ∈ E \ {0}. For all s > 0 define ψ(s) = Φ(sv). Clearly, ψ ∈ C 1 (R+ ) and ψ 0 (s) = hΦ0 (sv), vi, so that ψ 0 (s) = 0 if and only if sv ∈ N . The positive homogeneity of I0 , assumed in (v), gives ψ 0 (s) = hI00 (sv), vi − hI 0 (sv), vi = psp−1 I0 (v) − hI 0 (sv), vi.
26
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
Hence, ψ 0 (s) = 0 if and only if pI0 (v) = s1−p hI 0 (sv), vi,
(5.3)
since I0 (v) > 0 by (5.1). Now, by (ii), (iii), and the L’Hˆopital rule, the function s 7→ s1−p hI 0 (sv), vi is strictly increasing in R+ and maps R+ onto R+ . Hence, there exists a unique value sv > 0 satisfying (5.3). Moreover, for all s > 0 I(sv) ψ(s) = sp I0 (v) − p , (5.4) s and so ψ(s) ∼ sp I0 (v) as s → 0+ , since I(sv)/ksvkp → 0 by (i). Thus, ψ(s) > 0 for s > 0 close to zero, since I0 (v) > 0 by (5.1). On the other hand, ψ(s) → −∞ as s → ∞ by (iii). Therefore, the unique critical point sv ∈ R+ of ψ is a strict maximum point and satisfies (5.3) and so (a). (b) Take now (vn )n ⊂ E, with vn → v ∈ E \ {0} as n → ∞. The corresponding sequence (sn )n = (η(vn ))n is bounded in R+ . Otherwise, there would exist a subsequence, still denoted (sn )n , converging to ∞ as n → ∞. Clearly, (vn )n belongs to a weakly compact set W ⊂ E \ {0}, since it is bounded and E is a reflexive Banach space. Hence, ψ(sn ) → −∞ as n → ∞, by (5.4) and (iii). On the other hand, ψ(sn ) ≥ 0 for all n and in turn lim inf n→∞ ψ(sn ) ≥ 0, which is an obvious contradiction. Therefore, (sn )n is bounded and so admits a subsequence (snk )k converging to some s ∈ R+ 0 as k → ∞. The regularity of ψ forces that 0 = ψ 0 (snk ) → ψ 0 (s) = 0. The first part of the lemma implies that s = sv , and so sv > 0. This also shows that the entire sequence (sn )n converges to sv as n → ∞. Hence, η(vn ) = sn → sv = η(v) as n → ∞; that is, η is continuous. (c) By the previous argument the map m : z 7→ sz z = η(z)z = u is continuous from SE onto N , with continuous inverse m−1 : u 7→ u/kuk = z. (d) We claim that there exists δ > 0 such that sz ≥ δ for all z ∈ SE . Indeed, by (5.1) it follows that inf z∈SE I0 (z) ≥ c0 > 0 and so by (5.3), for all z ∈ SE , we get 0 s1−p z hI (sz z), zi = pI0 (z) ≥ pc0 > 0.
(5.5)
On the other hand, (i) implies that s1−p hI 0 (sz), zi → 0 as s → 0+ , and this fact, combined with (5.5), proves the claim. In particular, by virtue of the homeomorphism m between SE and N , for every u ∈ N we have that u = kuku/kuk = sz z, with z ∈ SE and sz uniquely determined by (a). Therefore, inf u∈N kuk = inf z∈SE ksz zk ≥ δ > 0; that is, (d) holds.
Existence theorems for eigenvalue problems
27
(e) Put c1 = inf v∈E\{0} maxs>0 Φ(sv) and c2 = inf z∈SE maxs>0 Φ(sz). Fix v ∈ E \ {0}. There exists a unique sv ∈ R+ satisfying (a). Since sv v ∈ N , clearly Φ(sv v) ≥ c, that is, maxs>0 Φ(sv) ≥ c. Hence, c1 ≥ c. Clearly, c1 ≤ c2 , because SE ⊂ E \ {0}. While the homeomorphism m between SE and N gives c2 = c. In conclusion, c ≤ c1 ≤ c2 = c, as desired. (f ) This follows immediately from properties (a) and (e). The lemma below is a slight modification of Proposition 14 of [38]; cf. Remark 15 of [38]. Lemma 5.2. The following properties hold. (a) If (un )n ⊂ N and supn Φ(un ) < ∞, then there exist u 6= 0 and a number su > 0 such that, up to a subsequence, su u ∈ N , un * u in E, and Φ(su u) ≤ lim inf Φ(un ); n→∞ (b) Φ N is coercive; that is, Φ(vn ) → ∞ as n → ∞ along any sequence (vn )n ⊂ N , with kvn k → ∞; (c) Φ satisfies the Palais–Smale condition on N . Proof. (a). Let (un )n ⊂ N be such that d = supn Φ(un ) < ∞. We first claim that (un )n is bounded in E. Assume for the sake of contradiction that, up to a subsequence, 0 < kun k → ∞ as n → ∞ and vn * v in E, vn = un /kun k, the space E being reflexive. We claim that v 6= 0. Suppose for the sake of contradiction that v = 0. By (iv) and Corollary 41.9 of [40] the functional I is weakly continuous in E, the space E being reflexive. Hence, I(svn ) → I(0) = 0 as n → ∞ by (v). Since vn ∈ SE and kun kvn = un ∈ N , we get that svn = kun k by Lemma 5.1-(a), svn being uniquely determined. Thus, for all s > 0 we have d ≥ Φ(un ) = Φ(svn vn ) ≥ Φ(svn ) ≥ c0 sp − I(svn ) → c0 sp as n → ∞ by (v). This yields a contradiction choosing s > (d/c0 )1/p . Hence v 6= 0, as claimed. It is easy to see from the proof of Lemma 5.1-(a) that Φ(sv v) > 0 for all v ∈ E \ {0} and so by Lemma 5.1-(e) c = inf Φ(u) = u∈N
inf
max Φ(sv) ≥ 0.
v∈E\{0} s>0
In particular, Φ(un ) ≥ c ≥ 0 for all n. Therefore, for n sufficiently large, 0≤
I(kun kvn ) Φ(un ) ≤ C0 − → −∞ p kun k kun kp
28
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
as n → ∞ by (iii) and (v), with W = {vn : n ∈ N} ∪ {v} ⊂ E \ {0}. This is again a contradiction. Therefore, (un )n is bounded and there exists u ∈ E such that un * u in E, up to a subsequence, E being reflexive. Assume for the sake of contradiction that u = 0. As before, by Lemma 5.1-(a) and the fact that un ∈ N , we get for all s > 0 d ≥ Φ(un ) ≥ Φ(sun ) ≥ d0 sp − I(sun ) → d0 sp > 0, where d0 = c0 inf u∈N kukp > 0 by (v) and Lemma 5.1-(d). Choosing s large enough, we find again a contradiction. Hence, u 6= 0, and we denote by su the unique number determined by Lemma 5.1-(a). Now, I(su un ) → I(su u) and J0 (su un ) → J0 (su u), by (iv) and (vi). Thus, Φ(su u) = I0 (su u) + J0 (su u) − I(su u) ≤ lim inf I0 (su un ) + lim J0 (su un ) − lim I(su un ) n→∞
n→∞
n→∞
= lim inf Φ(su un ) ≤ lim inf Φ(un ). n→∞
n→∞
(b) Let (vn )n ⊂ N be such that kvn k → ∞ as n → ∞. Suppose for the sake of contradiction that (Φ(vn ))n is bounded, up to a subsequence, since Φ(vn ) ≥ c. Then, by (a) of this lemma, there exists v ∈ E \ {0} such that, up to a further subsequence, vn * v as n → ∞. In particular, (kvn k)n is bounded. This is impossible. (c) Let (un )n ⊂ N be such that (Φ(un ))n is bounded and Φ0 (un ) → 0 as n → ∞. By (a) of this lemma, there exists u ∈ E \ {0} such that un * u in E as n → ∞. Consequently, I 0 (un ) → I 0 (u) and J00 (un ) → J00 (u) in E ? by (iv) and (vi), and in turn hI00 (un ) − I00 (u), un − ui = hΦ0 (un ) − Φ0 (u), un − ui − hJ00 (un ) − J00 (u), un − ui + hI 0 (un ) − I 0 (u), un − ui = o(1) as n → ∞. Hence, by the main inequality in (vi), it follows that kun k → kuk as n → ∞, and so un → u in E, the Banach space E being uniformly convex. Since we are going to argue on the function Ψ : SE → R, defined by Ψ(z) = Φ(u), where u = sz z is uniquely determined in Lemma 5.1-(a) and (c), let us recall the notion of weak slope, taken from Definition 2.1 of [12], together with some useful facts. Let (M, d) be a metric space, Ψ : M → R a continuous function, and z ∈ M . We denote by |dΨ|(z) the supremum of the real numbers σ in R+ 0
Existence theorems for eigenvalue problems
29
for which there exist δ > 0 and a continuous map H : Bδ (z) × [0, δ] → M such that for every v ∈ Bδ (z) and for every t ∈ [0, δ] it results that d(H(v, t), v) ≤ t,
Ψ(H(v, t)) ≤ Ψ(v) − σt.
(5.6)
The extended real number |dΨ|(z) is called the weak slope of Ψ at z. Proposition 5.3. Let M = (M, d) be a complete metric space. If the function Ψ : M → R is bounded below, then for all ε > 0 there exists zε ∈ M , with the property that Ψ(zε ) ≤ inf Ψ(z) + ε2 z∈M
and
|dΨ|(zε ) ≤ ε.
Proof. By Ekeland’s variational principle, for all ε > 0 there exists zε ∈ M such that Ψ(zε ) ≤ inf z∈M Ψ(z) + ε2 and Ψ(z) > Ψ(zε ) − εd(z, zε ) for all z ∈ M , with z 6= zε .
(5.7)
We claim that |dΨ|(zε ) ≤ ε. Otherwise there would exist δ > 0, σ > ε, and a continuous function H : Bδ (zε ) × [0, δ] → M such that (5.6) holds for every v ∈ Bδ (zε ) and t ∈ [0, δ]. Putting z = H(zε , δ) in the second inequality of (5.6), we get Ψ(z) ≤ Ψ(zε ) − σδ ≤ Ψ(zε ) − εd(z, zε ), which contradicts (5.7), since z 6= zε by (5.6). Let M denote a C 1 Finsler manifold. For the definition and the main properties of M we refer to [29, Section 2 and the theorem on page 201] and [28, Corollary at page 120]. Let Ψ ∈ C 1 (M ). Then Ψ0 (z) is an element of Tz (M )? . We say that z ∈ M is a critical point for Ψ if Ψ0 (z) = 0 and, in this case, c = Ψ(z) is called a critical value for Ψ. Moreover, kΨ0 (z)k?z =
sup hΨ0 (z), vi. v∈Tz (M ) kvkz =1
Proposition 5.4 (Corollary 2.12 of [12]). Let M be a C 1 Finsler manifold and Ψ : M → R be a C 1 function. Then for every z ∈ M , we have |dΨ|(z) = kΨ0 (z)k?z . The functional Ψ is said to satisfy the (P S)c condition, (P S)c for short, if for every sequence (zn )n ⊂ M such that Ψ(zn ) → c and kΨ0 (zn )k?zn → 0 as n → ∞, there exists a subsequence of (zn )n converging in M . Using Propositions 5.3 and 5.4, we now extend to complete Finsler manifolds Proposition 5.1 of [17], due to Ekeland for complete Riemannian manifolds.
30
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
Proposition 5.5. Let M be a complete C 1 Finsler manifold. If Ψ ∈ C 1 (M ) is bounded below, then for all ε > 0 there exists zε ∈ M such that Ψ(zε ) ≤ inf Ψ(z) + ε2 z∈M
and
kΨ0 (zε )k?zε ≤ ε.
In particular, if Ψ satisfies the (P S)c condition at c = inf Ψ(z), then c is a z∈M
critical value. Now we are in a position to present a slight modification of Theorem 13 of [38]. Theorem 5.6. Assume that E is a real infinite-dimensional uniformly convex Banach space and that SE is a complete C 1 Finsler manifold. Under the main assumptions (i)–(vi) of the section, equation Φ0 (u) = 0 admits a ground-state solution u; that is, Φ(u) = c. Moreover, if Φ is even, then the equation Φ0 (u) = 0 has infinitely many pairs of solutions. Proof. Define the function Ψ : SE → R by Ψ(z) = Φ(u), where u = sz z ∈ N is given by Lemma 5.1-(a) and (c). Clearly, Ψ0 (z) = sz Φ0 (u) in Tz (SE ) for each z ∈ SE by Lemma 5.1-(c). Hence, kΨ0 (z)k?z ≤ sz kΦ0 (u)kE ? , where sz = kuk ≥ δ > 0 by Lemma 5.2-(e). Furthermore, as shown in the proof of Corollary 10-(b) of [38], there exists a constant C ≥ 1, independent of z ∈ SE , such that kΨ0 (z)k?z ≤ kuk · kΦ0 (u)kE ? ≤ CkΨ0 (z)k?z for all z = m−1 (u) ∈ SE . Now, if (zn )n ⊂ SE is a (P S) sequence for Ψ, then n 7→ un = m(zn ) = szn zn ∈ N is a (P S) sequence for Φ by Lemma 5.1-(d). Therefore, by Lemma 5.2-(c) there exists a subsequence, say (un )n , such that un → u ∈ N in E, Φ being of class C 1 (E) and N closed. Hence, the corresponding subsequence zn → m−1 (u), as claimed. The value c = inf z∈SE Ψ(z) is critical by Ekeland’s variational principle given in Proposition 5.5; that is, there exists a minimizer zc ∈ SE , with Ψ(zc ) = c and Ψ0 (zc ) = 0. Therefore, uc = m(zc ) ∈ N is a ground-state solution for the equation Φ0 (u) = 0. Assume finally that Φ is even, so that also Ψ is even. Moreover, Ψ is bounded from below, since inf z∈SE Ψ(z) = c > 0 by Lemma 5.1-(e) and (f ). We claim that Ψ has infinitely many pairs of critical points. Indeed, put for each k = 1, 2, . . . Ek = {C ⊂ SE : C is compact and symmetric, i(C) ≥ k}, where i is the cohomological index for Z2 -action introduced by Fadell and Rabinowitz; see [18], [30], and also Section 2. By Proposition 2.5-(ii) and (iii),
Existence theorems for eigenvalue problems
31
the values ck = inf sup Ψ(z) → ∞ as k → ∞, C∈Ek z∈C
and ck are also critical, since (P S) holds, i(SE ) = dim E = ∞, and Ek 6= ∅ for all k. Hence, Ψ0 (z) = 0 has infinitely many pairs of solutions, as claimed. Thus, Φ0 (u) = 0 has infinitely many pairs of solutions, and the proof is completed. 6. Preliminary results for the Szulkin–Weth case We now turn back to the main problem (1.1) and assume throughout the section that the Carath´eodory functions g : Ω × R → R and h : ∂Ω × R → R satisfy conditions (g1 ) and (h1 ) of Section 4, while conditions (g2 )–(g4 ) and (h2 )–(h4 ) are replaced by G(x, s) g(x, s) = 0 and lim = ∞, uniformly in Ω \ N for (g2 )0 lim s→0 g0 (x)|s|p−1 |s|→∞ g0 (x)|s|p some N, with meas(N) = 0; g(x, s) (g3 )0 the function s 7→ is increasing in R− and in R+ for a.a. g0 (x)|s|p−1 x ∈ Ω; h(x, s) H(x, s) (h2 )0 lim = 0 and lim = ∞, uniformly in ∂Ω \ N s→0 h0 (x)|s|p−1 |s|→∞ h0 (x)|s|p for some N, with measN −1 (N) = 0; h(x, s) 0 (h3 ) the function s 7→ is increasing in R− and in R+ for a.a. h0 (x)|s|p−1 x ∈ ∂Ω. Problem (1.1) always admits the trivial solution u ≡ 0. Furthermore, (g3 )0 and (h3 )0 yield that g(x, s)s ≥ 0 for a.a. x ∈ Ω and h(x, s)s ≥ 0 for a.a. x ∈ ∂Ω and for all s ∈ R. Hence (g4 ) and (h4 ) of Section 4 are automatic. By the de l’Hˆ opital rule, (g2 )0 implies (g2 ), and so by (g1 ) G(x, s) = 0, uniformly in Ω \ O, s→0 w2 (x)|s|p lim
with meas(O) = 0. Similarly, (h2 )0 gives (h2 ), and in turn by (h1 ) H(x, s) = 0, s→0 w3 (x)|s|p lim
uniformly in ∂Ω \ O for some O, with measN −1 (O) = 0.
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
In order to prove the main results in the framework of this section, we put 1 λ Φ(u) = I0 (u) − I(u), I0 (u) = I0 (u) + J0 (u) = kukp − kukpp,f , p p I(u) = I1 (u) + I2 (u), (6.1) Z Z H(x, u(x)) dS. G(x, u(x)) dx, I2 (u) = I1 (u) = Ω
∂Ω
Obviously, I0 , I0 , J0 , I1 , I2 , and I are of class C 1 (E) and Z Z p−2 0 b(x)|u|p−2 uv dS a(x)|∇u| ∇u · ∇v dx + hI0 (u), vi = ∂Ω Ω Z − λ f (x)|u|p−2 uv dx,
(6.2)
Ω
hI 0 (u), vi = hI10 (u), vi + hI20 (u), vi, Z 0 hI1 (u), vi = g(x, u(x))v(x) dx, Ω
hI20 (u), vi
Z =
h(x, u(x))v(x) dS ∂Ω
for every u, v ∈ E. Thus, the solutions of (1.1) coincide with the critical points of Φ. Moreover, I0 is positively homogeneous of degree p, and for all λ ∈ R 1 λ− I0 (u) ≤ 1− kukp , (6.3) p λ1 while if λ < λ1 we also have 1 λ+ I0 (u) ≥ 1− kukp , (6.4) p λ1 where λ+ = max{λ, 0} and λ− = min{λ, 0}. Furthermore, I0 is weakly lower semicontinuous in E, being convex, and I00 satisfies hI00 (u) − I00 (v), u − vi ≥ (kukp−1 − kvkp−1 ) (kuk − kvk)
(6.5)
for all u, v ∈ E by Lemma 3.2. Proposition 6.1. J00 is completely continuous from E into E ? and J0 is weakly continuous in E. Proof. It is enough to show that J00 maps weakly convergent sequences of E into strongly convergent sequences of E ? . Let (un )n ⊂ E be such that un * u in E. Hence un → u in Lp (Ω, f ), since the embedding E ,→ Lp (Ω, f ) is compact by (2.1). Denote by C > 0 a constant such that kϕkp,f ≤ Ckϕk for all ϕ ∈ E.
Existence theorems for eigenvalue problems
33
In particular, kun kp,f → kukp,f , or equivalently, kvn kp0 ,f → kvkp0 ,f , where 0 vn = |un |p−2 un and similarly v = |u|p−2 u. We claim that vn → v in Lp (Ω, f ). Indeed, fix any subsequence (vnk )k of (vn )n . The related subsequence (unk )k of (un )n converges in Lp (Ω, f ) and admits a subsequence, say (unkj )j , converging to u a.e. in Ω. Hence, the corresponding subsequence (vnkj )j of (vnk )k converges to v a.e. in Ω. Therefore, since 1 < p0 < ∞, by the Clark0 son and Mil’man theorems it follows that vnkj * v in Lp (Ω, f ), since the sequence (kvn kp0 ,f )n is bounded, and so by Radon’s theorem we get that 0 vnkj → v in Lp (Ω, f ), since kvn kp0 ,f → kvkp0 ,f . This shows the claim, since the subsequence (vnk )k of (vn )n is arbitrary. For all φ ∈ E, with kφk = 1, we have Z 0 0 |hJ0 (un ) − J0 (u), φi| ≤ λ f (x)|vn − v| · |φ| dx ≤ λkvn − vkp0,f kφkp,f Ω
≤ Ckvn − vkp0,f by H¨ older’s inequality. In other words, kJ00 (un ) − J00 (u)kE ? → 0 as n → ∞. Thus, J0 is weakly continuous in E by Corollary 41.9 of [40], the space E being reflexive. Lemma 6.2. The functions g(x, s) , g(x, s) = w2 (x)|s|p−1 0, G(x, s) , G0 (x, s) = w2 (x)|s|p 0, h(x, s) , h(x, s) = w3 (x)|s|p−1 0, H(x, s) , H0 (x, s) = w3 (x)|s|p 0,
x ∈ Ω, s 6= 0, x ∈ Ω, s = 0, x ∈ Ω, s 6= 0, x ∈ Ω, s = 0, x ∈ ∂Ω, s 6= 0, x ∈ ∂Ω, s = 0, x ∈ ∂Ω, s 6= 0, x ∈ ∂Ω, s = 0,
are of Carath´eodory type, and the Nemytskii operators Ng , NG0 : Lr (Ω, w2 ) → Lr/(r−p) (Ω, w2 ), Nh , NH0 : Lq (∂Ω, w3 ) → Lq/(q−p) (∂Ω, w3 )
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
are continuous and bounded, with Ng (0) = NG0 (0) = Nh (0) = NH0 (0) = 0. Proof. By (g1 ) and (g2 )0 along with (h1 ) and (h2 )0 it is clear that g, G0 , h, and H0 are well-defined and of Carath´eodory type. Moreover, they also imply that g0 (x) + |s|r−p ) |g(x, s)| ≤ Cg (˜ g0 (x) + |s|r−p ) and 0 ≤ G0 (x, s) ≤ CG0 (˜ for a.a. x ∈ Ω, and ˜ 0 (x) + |s|q−p ) ˜ 0 (x) + |s|q−p ) and 0 ≤ H0 (x, s) ≤ CH (h |h(x, s)| ≤ Ch (h 0 for a.a. x ∈ ∂Ω, where Cg , CG0 , Ch , and CH0 are suitable positive con˜ 0 = h0 /w3 . By (g1 ) and (h1 ) we have stants and g˜0 = g0 /w2 , while h r/(r−p) ˜ g˜0 ∈ L (Ω, w2 ) and h0 ∈ Lq/(q−p) (∂Ω, w3 ), so that the result follows at once by applying Lemma 2.2-(i), with A = Ω, d = N , w = w2 , α = r, and β = r/(r − p), and Lemma 2.2-(i), with A = ∂Ω, d = N − 1, w = w3 , α = q, and β = q/(q − p). kI 0 (u)kE ? I(u) = 0 and lim = 0. p−1 kuk→0 kuk kuk→0 kukp Proof. We first show that kI10 (u)kE ? = o kukp−1 as kuk → 0. Fix u ∈ E, with kuk > 0. Using the notation of Lemma 6.2, we have Z kI10 (u)kE ? 1−p ≤ sup kuk |g(x, u(x))| · |ϕ(x)| dx = sup kuk1−p J(u, ϕ), kukp−1 Ω kϕk=1 kϕk=1 Lemma 6.3. We have lim
where Z J(u, ϕ) =
|g(x, u(x))|w2 (x)|u(x)|p−1 |ϕ(x)| dx.
Ω
By H¨ older’s inequality, we obtain for all ϕ ∈ E, with kϕk = 1, Z (r−p)/r J(u, ϕ) ≤ w2 (x)|g(x, u(x))|r/(r−p) dx kukp−1 r,w2 kϕkr,w2 Ω Z (r−p)/r ≤C w2 (x)|g(x, u(x))|r/(r−p) dx kukp−1 , Ω
where C is a suitable constant related to the Sobolev constant of the continuous embedding E ,→ Lr (Ω, w2 ). In conclusion, we have shown that Z (r−p)/r kI10 (u)kE ? r/(r−p) ≤ C w (x)|g(x, u(x))| dx →0 2 kukp−1 Ω
Existence theorems for eigenvalue problems
35
as kuk → 0 by Lemma 6.2. Furthermore, (r−p)/r Z |I1 (u)| r/(r−p) w (x)|G (x, u(x))| dx →0 ≤ C 2 0 kukp Ω as kuk → 0, again by Lemma 6.2. Similarly, kI20 (u)kE ? = o kukp−1 and I2 (u) = o (kukp ) as kuk → 0. This completes the proof. Lemma 6.4. The following properties hold. (α) s 7→ s1−p hI 0 (su), ui is strictly increasing for u 6= 0 and s > 0; (β) I 0 is completely continuous in E. Proof. (α) follows directly from (g3 )0 and (h3 )0 . (β) Condition (g1 ) implies that |g(x, s)| ≤ g0 (x) + Kg |s|r−1 for a.a. x ∈ Ω and s ∈ R, where Kg = (1 + Cg )Cg . Hence, Lemma 2.2-(i), with A = Ω, d = N , w = w2 , α = r, and β = r0 , imply that the Nemytskii operator Ng 0 associated to g is bounded and continuous from Lr (Ω, w2 ) to Lr (Ω, w2 ), 0 since g0 ∈ Lr (Ω, w2 ). Moreover, I10 can be viewed as the composition Ng
0
I10 : E ,→ Lr (Ω, w2 ) → Lr (Ω, w2 ),→E ? , where E ,→ Lr (Ω, w2 ) is completely continuous by Theorem 1 of [31] and (g1 ). Hence, I10 is completely continuous in E, being a composition of continuous operators, with the first completely continuous. In the same way, we prove that I20 is completely continuous by virtue of (h1 ). Indeed, condition (h1 ) forces that |h(x, s)| ≤ h0 (x) + Kh |s|q−1 for a.a. x ∈ ∂Ω and all u ∈ R, with Kh = (1 + Ch )Ch , so that Lemma 2.2-(i) can be applied with A = ∂Ω, d = N − 1, w = w3 , α = q, and β = q 0 , 0 since h0 ∈ Lq (∂Ω, w3 ) by (h1 ). In conclusion, the operator I 0 is completely continuous in E. Lemma 6.5. lims→∞ s−p I(su) = ∞ uniformly for u on weakly compact subsets of E \ {0}. Proof. Let W ⊂ E \ {0} be a weakly compact subset of E \ {0}, and fix (un )n ⊂ W. It is enough to show that if sn → ∞ as n → ∞, then there exists a subsequence of (s−p n I(sn un ))n which diverges to ∞. Now, there exists a subsequence of (un )n , still labeled (un )n , and u ∈ E \ {0}, such that un * u in E,
un → u in Lp (Ω, w1 ), p
un → u in L (∂Ω, w4 ), w4 (x) = (1 + |x|)
−α4
un → u a.e. in Ω,
un → u a.e. on ∂Ω,
, x ∈ ∂Ω,
p − 1 < α4 < N,
(6.6)
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
by Theorem 1 of [31], where w1 is defined in (2.1). Hence, either |u| > 0 in a subset A of Ω, with meas(A) > 0, or |u| > 0 in a subset Γ of ∂Ω, with measN −1 (Γ) > 0, since kuk > 0 and w1 > 0 in Ω, and w4 > 0 in ∂Ω Case |u| > 0 in A, meas(A) > 0. For n sufficiently large G(x, sn un (x)) G(x, sn un (x)) g0 (x)|un (x)|p = p g0 (x)|sn un (x)|p sn
a.e. in A.
Hence, s−p n G(x, sn un (x)) → ∞ a.e. in A by (g2 ), since |sn un (x)| → ∞ as n → ∞. Thus, by (g4 ), which continues to hold, and the Fatou lemma, Z I1 (sn un ) G(x, sn un (x)) = g (x)|un (x)|p dx → ∞ as n → ∞. p p 0 g (x)|s u (x)| sn 0 n n Ω Case |u| > 0 in Γ, measN −1 (Γ) > 0. Similarly, using (h2 )0 in place of (g2 )0 and arguing as above, by (h4 ) we obtain Z H(x, sn un ) I2 (sn un ) = h (x)|un (x)|p dx → ∞ as n → ∞. p 0 h (x)|s u (x)| spn n n ∂Ω 0 This shows that s−p n I(sn un ) → ∞ as n → ∞ and completes the proof, (sn )n being an arbitrary sequence diverging to ∞. 7. Existence in the Szulkin–Weth case We are going to apply the main results of Section 5 to problem (1.1) under the assumptions of Section 6, with E now the real infinite-dimensional uniformly convex Banach space defined in Section 2, and the Euler–Lagrange functional Φ associated to (1.1), given in (4.1), being thought of as in (6.1). Hence Ψ(z) = Φ(u), u = m(z), for all z ∈ SE , that is, when M = SE . Of course SE is a complete metric space. Consider g(t) = tp−1 , which is continuous and strictly increasing in R+ 0, and the corresponding duality map Jg , defined by Jg (u) = {u? ∈ E ? : hu? , ui = g(kuk)kuk = kukp , ku? kE ? = g(kuk) = kukp−1 } for all u ∈ E. The functional G : E → R, given by Z kuk 1 G(u) = g(t) dt = kukp , p 0
Existence theorems for eigenvalue problems
37
is of class C 1 (E), and so G 0 (u) = Jg (u) for all u ∈ E by Theorem 1-(iii) of [15]; that is, for all u, v ∈ E Z Z hG 0 (u), vi = a(x)|∇u|p−2 ∇u∇v dx + b(x)|u|p−2 uv dS = hJg (u), vi. Ω
∂Ω
In particular, Jg : E → E ? is a single-valued continuous map. Furthermore, Jg maps bounded sets of E into bounded sets of E ? and SE = {z ∈ E : hJg (z), zi = 1}. Hence SE is a complete C 1 manifold of E. In particular, the tangent space Tz (SE ) of SE at z ∈ SE can be written in the form Tz (SE ) = {v ∈ E : hJg (z), vi = 0} = KerJg (z). Thus, SE has a Finsler structure on the bundle T (SE ); see [29, Section 2] and [28, Corollary at page 120]. In other words, SE is a complete C 1 Finsler manifold. As an application of Theorem 5.6 we are now able to prove the main result of the section. Theorem 7.1. Suppose that (g1 ), (g2 )0 –(g3 )0 , (h1 ), and (h2 )0 –(h3 )0 hold and that λ < λ1 . Then we have the following: (a) Problem (1.1) has a ground-state solution, that is, a nontrivial solution of minimal energy. Moreover, if g and h are odd in u, then (1.1) has infinitely many pairs of solutions. (b) Problem (1.1) admits a least-energy sign-changing solution. Proof of Theorem 7.1-(a). Whenever λ < λ1 , the functional Φ = I0 − I, defined in (6.1), satisfies all the assumptions (i)–(vi) of Theorem 5.6. Indeed, Lemmas 6.3–6.5 assure the validity of (i)–(iv). Moreover, I0 is trivially positively homogeneous of degree p and the main inequality in (v) is satisfied with c0 = (1 − λ+ /λ1 )/p > 0 and C0 = (1 − λ− /λ1 )/p ≥ c0 ; cf. (6.3)–(6.4). Hence, also (v) holds. Finally, as noted in Section 6, also (vi) is satisfied by virtue of Lemma 3.2 and Proposition 6.1. Therefore, by Theorem 5.6 problem (1.1) admits a ground-state solution in E. Moreover, if g and h are odd in u, then Φ is even, and so by Theorem 5.6 problem (1.1) has infinitely many pairs of solutions in E. This completes the proof of Theorem 7.1-(a). In order to prove the existence of least-energy sign-changing solutions of problem (1.1), we put Nsc := {u ∈ E : u+ , u− ∈ N }
and csc := inf Φ(u). u∈Nsc
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Giuseppina Autuori, Patrizia Pucci and Csaba Varga
First observe that if u ∈ E, then u+ , u− ∈ E and Φ(u) = Φ(u+ ) + Φ(u− ). Moreover, if u is a solution of (1.1), taking v = u+ in (6.2), we obtain that 0 = hΦ0 (u), u+ i = hΦ0 (u+ ), u+ i and, similarly, taking v = u− in (6.2), we get 0 = hΦ0 (u), u− i = hΦ0 (u− ), u− i. Hence, u+ , u− ∈ N ; that is, every sign-changing solution u ∈ E of Φ0 (u) = 0 lies in Nsc . Proof of Theorem 7.1-(b). We claim that (1.1) admits a least-energy sign-changing solution u ∈ Nsc , with Φ(u) = csc ≥ 2c, where c is given in Lemma 5.1-(e). Let (un )n ⊂ Nsc be a minimizing sequence; that is, Φ(un ) → csc as n → ∞. − + − Now, Φ(un ) = Φ(u+ n ) + Φ(un ), with Φ(un ) ≥ c and Φ(un ) ≥ c for all n. − In particular, (Φ(u+ n ))n and (Φ(un ))n are bounded. Hence, by Lemma 5.2(a) there exists a subsequence, still labeled (un )n , such that u+ n * v 6= 0, + → v, and u− → z a.e. in Ω. Without loss of generality u− * z = 6 0, u n n n − we may also assume (Φ(u+ n ))n and (Φ(un ))n are converging. Therefore, − 0 = u+ n un → vz = 0 a.e. in Ω. Moreover, again by Lemma 5.2-(a), there exist sv > 0 and sz > 0 such that u = sv v + sz z = u+ + u− ∈ Nsc and − 2c ≤ Φ(u) = Φ(sv v) + Φ(sz z) ≤ lim inf Φ(u+ n ) + lim inf Φ(un ) n→∞
n→∞
= lim Φ(un ) = csc . n→∞
In particular, Φ(u) = csc ≥ 2c, as claimed. The final delicate step consists in proving that Φ0 (u) = 0, using the argument of the proof of Theorem 18 of [38], which we repeat here for the sake of clarity. For all s, t > 0, with at least one of them different from 1, by Lemma 5.1-(a) we have Φ(su+ + tu− ) = Φ(su+ ) + Φ(tu− ) < Φ(u+ ) + Φ(u− ) = Φ(u) = csc . (7.1) Assume for the sake of contradiction that Φ0 (u) 6= 0. Then, there exist δ > 0 and µ > 0 such that kΦ0 (v)kE ? ≥ µ for all v ∈ E, with kv − uk ≤ 3δ. Put φ(s, t) = su+ + tu− for all (s, t) ∈ D = I × I, I = [1/2, 3/2]. Clearly, the definition of φ and (7.1) imply that Φ(φ(s, t)) = csc if and only if s = t = 1 and Φ(φ(s, t)) < csc otherwise. In particular, cm = max Φ(φ(s, t)) < csc .
(7.2)
(s,t)∈∂D
The deformation Lemma 2.3 of [39], with ε = min{(csc − cm )/4, δµ/8} and S the closed ball of E with center u and radius δ, yields the existence of a continuous deformation ℘ : [0, 1] × E → E such that (α) ℘(1, v) = v for all v 6∈ Φ−1 ([csc − 2ε, csc + 2ε]);
Existence theorems for eigenvalue problems
39
(β) Φ(℘(1, v)) ≤ csc − ε for every v ∈ E, with kv − uk ≤ δ and Φ(v) ≤ csc + ε; (γ) Φ(℘(1, v)) ≤ Φ(v) for all v ∈ E. Property (γ) gives at once that max(s,t)∈D Φ(℘(1, φ(s, t))) < csc . The next step is to prove that ℘(1, φ(D)) ∩ Nsc 6= ∅, which contradicts the definition of csc and completes the proof. Define for all (s, t) ∈ D the continuous maps Ψ0 (s, t) = (hΦ0 (su+ ), u+ i, hΦ0 (tu− ), u− i), 1 Ψ1 (s, t) = hΦ0 (℘(1, φ(s, t))+ ), ℘(1, φ(s, t))+ i, s 1 0 hΦ (℘(1, φ(s, t))− ), ℘(1, φ(s, t))− i . t 1 + The C (R ) function ψ(s) = Φ(su+ ) admits a unique strict maximum point at su+ = 1, since ψ 0 (1) = hΦ0 (u+ ), u+ i = 0 and u+ ∈ N ; see the proof of Lemma 5.1-(a). Moreover, ψ 0 (s) = hΦ0 (su+ ), u+ i > 0 for s ∈ (0, 1), while ψ 0 (s) = hΦ0 (su+ ), u+ i < 0 for s > 1. In particular, the continuous function I 3 s 7→ hΦ0 (su+ ), u+ i is positive if s = 1/2 and negative if s = 3/2, so that the formula (3.15) of Proposition 3.4.1 in [16] implies that deg(hΦ0 (su+ ), u+ i, I, 0) = −1. Similarly, using the above argument, with u− in place of u+ , we get that the continuous function I 3 t 7→ hΦ0 (tu− ), u− i is positive if t = 1/2 and negative if t = 3/2, so that again deg(hΦ0 (tu− ), u− i, I, 0) = −1. The cartesian product property in Theorem 3.16-7 of [36] gives that deg(Ψ0 , D, (0, 0)) = 1. By (7.2) and (α) we deduce that φ(s, t) = ℘(1, φ(s, t)) for all (s, t) ∈ ∂D. Thus, Ψ0 (s, t) = Ψ1 (s, t) for all (s, t) ∈ ∂D, and the dependence only on the boundary-values property in Theorem 3.16-2 of [36] yields deg(Ψ1 , D, (0, 0)) = deg(Ψ0 , D, (0, 0)) = 1. Therefore, Corollary 3.4.1. of [16] assures the existence of some (s, t) ∈ D such that Ψ1 (s, t) = (0, 0); that is, ℘(1, φ(s, t))+ , ℘(1, φ(s, t))− ∈ N ; in other words, ℘(1, φ(s, t)) ∈ Nsc . Hence, ℘(1, φ(D)) ∩ Nsc 6= ∅. This contradiction shows that Φ0 (u) = 0, as claimed. In the last part of the section we complete the investigation on the existence of solutions of (1.1), considering the case λ ≥ λ1 , under conditions (g1 ), (g2 )0 –(g3 )0 , (h1 ), and (h2 )0 –(h3 )0 . Let us start with a preliminary monotonicity result. Proposition 7.2. The function s 7→ G(x, s) = g(x, s)s − pG(x, s) is nondecreasing in R+ for a.a. x ∈ Ω. Similarly, for a.a. x ∈ ∂Ω the function s 7→ H(x, s) = h(x, s)s − pH(x, s) is non-decreasing in R+ .
40
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
Proof. Let 0 < s ≤ t. By (g3 )0 , for all x ∈ Ω \ N, meas(N) = 0, we have n1 o G(x, t) − G(x, s) = p [g(x, t)t − g(x, s)s] − [G(x, t) − G(x, s)] p Z s Z t n Z t g(x, t) g(x, s) p−1 g(x, τ ) p−1 o p−1 =p τ dτ − τ dτ − τ dτ tp−1 sp−1 τ p−1 0 0 s Z sh n Z t h g(x, t) g(x, τ ) i g(x, t) g(x, s) i p−1 o p−1 =p − τ dτ + − p−1 τ dτ tp−1 τ p−1 tp−1 s 0 s ≥ 0. Similarly, using assumption (h3 )0 in place of (g3 )0 , we obtain the analogous conclusion for the function H, as stated. Lemma 7.3. The functional Φ satisfies the (P S)` condition in E for all λ ≥ 0 and ` ∈ R. Proof. Let (un )n ⊂ E be a (P S)` sequence; that is, Φ(un ) → ` and Φ0 (un ) → 0 as n → ∞. First we claim that (un )n is bounded. Proceed by contradiction and suppose that 0 < kun k → ∞ as n → ∞, up to a subsequence. Hence, using the notation of Proposition 7.2, we find that Z Z G(x, un ) dx + H(x, un ) dS = pΦ(un ) − hΦ0 (un ), un i → p ` (7.3) Ω
∂Ω
as n → ∞. Define vn = un /kun k for all n. By Proposition A.2, (2.1), (g1 ), and (h1 ) there exists v ∈ E such that, up to a subsequence, vn * v in E, vn → v in Lr (Ω, w2 ),
vn → v in Lp (Ω, f ), vn → v in Lq (∂Ω, w3 ).
We first assume for the sake of contradiction that v = 0. Let sn be the smallest value of t ∈ [0, 1] such that Φ(tun ) = max Φ(sun ).
(7.4)
s∈[0,1]
√ Now, fix k ∈ N and define zn = p 2pk vn . Hence, zn → 0 in Lr (Ω, w2 ) as n → ∞. Similarly, also zn → 0 in Lq (∂Ω, w3 ) as n → ∞. Consequently, by (g4 ) and (h4 ) (which continue to hold), the H¨older inequality, and Lemma 6.2 Z Z (r−p)/r 0≤ G(x, zn ) dx ≤ w2 (x)G0 (x, zn )r/(r−p) dx kzn kpr,w2 = o(1), Ω Ω Z Z (q−p)/q 0 ≤ H(x, zn ) dS ≤ w3 (x)H0 (x, zn )q/(q−p) dS kzn kpq,w3 ,∂Ω = o(1) ∂Ω
∂Ω
Existence theorems for eigenvalue problems
41
as n → ∞. Therefore, by (7.4) and the fact kzn kp,f → 0, we get Z Z p H(x, zn ) dS ≥ k G(x, zn ) dx − Φ(sn un ) ≥ Φ(zn ) = 2k − λkzn kp,f − ∂Ω
Ω
for all n sufficiently large. That is, Φ(sn un ) → ∞ as n → ∞. Moreover, Φ(0) = 0 and Φ(un ) → ` ∈ R, so that sn ∈ (0, 1) for all n sufficiently large, and in turn Z o nZ p p p h(x, sn un )un dS g(x, sn un )un dx − sn (kun k − λkun kp,f ) − sn ∂Ω
Ω
d = hΦ (sn un ), sn un i = sn Φ(sun ) = 0. ds s=sn 0
Thus, by Proposition 7.2, it follows that Z Z Z Z G(x, un ) dx + H(x, un ) dS ≥ G(x, sn un ) dx + H(x, sn un ) dS Ω ∂Ω Ω ∂Ω Z Z = spn (kun kp − λkun kpp,f ) − p G(x, sn un ) dx + H(x, sn un ) dS Ω
∂Ω
= pΦ(sn un ) → ∞ as n → ∞, which contradicts (7.3). This means that the case v = 0 cannot occur. Let us then suppose v 6= 0. Since λ ≥ 0, condition (6.3) implies that I0 (u) ≤ kukp /p for all u ∈ E. Therefore, Φ(un ) 1 I(kun kvn ) ≤ − → −∞ p kun k p kun kp by Lemma 6.5, with W = {vn : n ∈ N} ∪ {v} ⊂ E \ {0}. Hence also the case v 6= 0 is impossible. In conclusion, the sequence (un )n must be bounded in E. Thus, by Proposition A.2, we have that un * u in E, up to a subsequence. By (6.2) hΦ0 (un ), un − ui = hI00 (un ), un − ui − hJ00 (un ), un − ui − hI 0 (un ), un − ui → 0, since (un )n is a (P S)` sequence. As shown in Proposition 6.1, the Fr´echet derivative J00 : E → E ? is completely continuous, and this implies in particular that hJ00 (un ), un − ui → 0 as n → ∞. Moreover, Lemma 6.4-(β) implies that hI 0 (un ), un − ui → 0 as n → ∞. Consequently, hI00 (un ), un − ui → 0 as n → ∞. Therefore, by (6.5) in particular kun k → kuk as n → ∞. By Proposition A.2 it follows at once that un → u as n → ∞ in E, as claimed.
42
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
We can follow the mini-max construction of Section 3. As shown in Proposition 3.4, the mini-max eigenvalues defined in (3.5)–(3.6), where now I = pI0 , are such that λ1 = min pI0 (u), u∈M
λk ∈ R+ ,
lim λk = ∞,
k→∞
i({u ∈ E \ {0} : I0 (u) ≤ λk kukpp,f }) = i({u ∈ E : I0 (u) < λk+1 kukpp,f }) = k for all k, with λk < λk+1 . In particular, λ1 coincides with the first eigenvalue of (3.1) given in (3.2). Theorem 7.4. Under (g1 ), (g2 )0 –(g3 )0 , (h1 ), and (h2 )0 –(h3 )0 , problem (1.1) admits a nontrivial solution for all λ ≥ λ1 . Proof. Since λ1 < λ2 ≤ · · · ≤ λk → ∞ and λ ≥ λ1 , there exists k ≥ 1 such that λk ≤ λ < λk+1 . Define the two symmetric closed cones o o n n C− = u ∈ E : I0 (u) ≤ λk kukpp,f , C+ = u ∈ E : I0 (u) ≥ λk+1 kukpp,f . Clearly, C− ∩C+ = {0} and i(C− \{0}) = i(E\C+ ) = k, as noted above. Furthermore, for all u ∈ C+ inequality (4.3) continues to hold. Now, Lemma 6.3 implies that kuk−p I(u) → 0 as kuk → 0. Therefore, there exists a number r+ > 0 such that for all u ∈ C+ , with kuk = r+ , we have Φ(u) ≥ α, where p α = r+ (1−λ/λk+1 )/2p > 0. On the other hand, Φ(u) ≤ (λk −λ)kukpp,f /p ≤ 0 for all u ∈ C− , since (g4 ) and (h4 ) are still valid. The embeddings E ,→ Lp (Ω, f ) and E ,→ Lp (∂Ω, w4 ) are compact by Theorem 1 of [31], where w4 is defined in (6.6). Hence, the cone C− is also closed in the real normed space E = (E, |||·|||), where |||·||| = k·kp,f +k·kp,w4 ,∂Ω . Taking e ∈ E \ C− and t > 0, we easily see, as in proof of Theorem 4.3, that (4.4) holds. We claim that there exists r− > r+ such that Φ(u) ≤ 0 for all u in C− + R+ e, with kuk ≥ r− . Otherwise, there would exist a sequence (un )n in C− + R+ e ⊂ E, with kun k → ∞, such that Φ(un ) > 0. In particular, 0 < kun k ≤ κ |||un ||| = κ (kun kp,f + kun kp,w4 ,∂Ω ). Hence, up to a subsequence, we have vn = un /kun k ∈ SE for all n and vn * v 6= 0, since 1 ≤ κ limn (kvn kp,f + kvn kp,w4 ,∂Ω ) = κ (kvkp,f + kvkp,w4 ,∂Ω ) = κ |||v|||. Put W = {vn : n ∈ N} ∪ {v} ⊂ E \ {0}. Then, 1 λ 0 < Φ(un ) ≤ 1+ kun kp − I(kun kvn ) ∼ −I(kun kvn ) → −∞ p λ1 as n → ∞ by Lemma 6.5. This contradiction proves the claim. The geometrical construction of Theorem 2.7 is completed, so that the corresponding sets satisfy the assertion. In particular, (Q, D− ∪ H) links
Existence theorems for eigenvalue problems
43
S+ cohomologically in dimension k + 1 over Z2 and inequalities (4.5) remain valid also in this setting. Finally, Φ ∈ C 1 (E) satisfies the (P S)` condition by Lemma 7.3 for all λ ≥ λ1 and ` ∈ R. Therefore, problem (1.1) admits a weak nontrivial solution u ∈ E, with Φ(u) ≥ α by virtue of Theorem 2.7. Appendix A Following some ideas contained in [4, Proposition A.9] and [15, Theorem 6], we show that the Banach space E = (E, k · k), defined in the Introduction, is uniformly convex. To do this, we first prove the validity of the Clarkson inequalities in the space RN endowed with the Euclidean norm | · |. The original proof of Clarkson in C can be found in [1, Lemma 2.27]. Lemma A.1. If 1 < p ≤ 2, then for all x, y ∈ RN x + y p0 x − y p0 1 1/(p−1) 1 |x|p + |y|p . (A.1) + ≤ 2 2 2 2 If p ≥ 2 then for all x, y ∈ RN x + y p x − y p 1 1 (A.2) + ≤ |x|p + |y|p . 2 2 2 2 Proof. First observe that (A.1) and (A.2) coincide when p = 2, and in this case they hold with the equality sign, being exactly the parallelogram law. On the other hand, they also hold with the equality sign if either x = 0 or y = 0. Hence, in the sequel of the proof, we consider only the nontrivial case. Let 1 < p < 2. Condition (A.1) is equivalent to 0 0 1/p0 0 |x + y|p + |x − y|p ≤ 21/p (|x|p + |y|p )1/p . (A.3) Let us distinguish two cases. Case |x| ≥ |y| > 0. Put for convenience w = x/|x| and v = y/|x|, so that |w| = 1 and 0 < |v| ≤ 1. Dividing (A.3) by |x| > 0, we get 0
0
|w + v|p + |w − v|p ≤ 2(1 + |v|p )1/(p−1) . (A.4) PN For brevity, let a = i=1 wi vi , so that |a| ≤ c ≤ 1, where c = |v| ∈ (0, 1]. Thus, (A.4) is equivalent to 0
0
(1 + 2a + c2 )p /2 + (1 − 2a + c2 )p /2 ≤ 2(1 + cp )1/(p−1) . The function 0
0
fc (a) = (1 + 2a + c2 )p /2 + (1 − 2a + c2 )p /2
(A.5)
44
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
is well–defined and non–negative in [−c, c]. Moreover, fc is even and admits 0 0 its maximum value in a = ±c, with fc (±c) = (1 + c)p + (1 − c)p . Hence, (A.5) reduces to 0
0
(1 + c)p + (1 − c)p ≤ 2(1 + cp )1/(p−1) , which is exactly the original formula (7) of Theorem 2 of Clarkson in [8]. Case 0 < |x| < |y|. Due to the symmetry of (A.3), we can repeat the proof of the first case, interchanging x with y, so that now v = x/|y| and |v| ∈ (0, 1). Let p > 2. Clearly 1 < p0 < 2, and so (A.1) becomes for p0 x + y p x − y p 1 1 0 1 1 0 0 p/p |x|p + |y|p + ≤ |x|p + |y|p ≤ 2 2 2 2 2 2 0 2p/p −1 ≤ p/p0 (|x|p + |y|p ), 2 that is (A.2) holds. Proposition A.2. The Banach space E = (E, k · k), defined in the Introduction, is uniformly convex. Proof. Let us first consider the case p ≥ 2. Fix ε ∈ (0, 2) and let u, v ∈ E be such that kuk = kvk = 1 and ku − vk ≥ ε. Using Lemma A.1, we have
u + v p u − v p
+
2 2 Z Z n ∇u + ∇v p ∇u − ∇v p o n u + v p u − v p o = a(x) b(x) + dx+ + dS 2 2 2 2 Ω ∂Ω Z Z n o n o 1 1 a(x) |∇u|p + |∇v|p dx + b(x) |u|p + |v|p dS ≤ 2 Ω 2 ∂Ω 1 = kukp + kvkp = 1. 2 Therefore,
u + v p ε p
,
≤1− 2 2 and so, taking δ = δ(ε) such that 1 − (ε/2)p = (1 − δ)p , we obtain that ku + vk ≤ 2(1 − δ). Suppose now 1 < p < 2. Fix ε ∈ (0, 22/p ) and let u, v ∈ E be such that kuk = kvk = 1 and ku − vk ≥ ε. First note that for any ϕ ∈ E, we have that 0 |∇ϕ|p ∈ Lp−1 (Ω, a), since Z 1 0 0 p−1 k|∇ϕ|p kp−1,a = a(x)|∇ϕ|p (p−1) dx Ω
Existence theorems for eigenvalue problems
=
Z
a(x)|∇ϕ|p dx
1 p−1
45 0
= k∇ϕkpp,a .
Ω
Hence, the reverse Minkowski inequality proved in Proposition A.5 of [4] yields
∇u − ∇v p0
∇u + ∇v p0
+
2 2 p,a p,a
∇u + ∇v p0
∇u − ∇v p0
= + 2 2 p−1,a p−1,a
∇u + ∇v p0 ∇u − ∇v p0
≤ + 2 2 p−1,a Z h ∇u + ∇v p0 ∇u − ∇v p0 p−1 i1/(p−1) = a(x) dx + 2 2 Ω 1/(p−1) 1 p p k∇ukp,a + k∇vkp,a ≤ , 2 where in the last step we have used the inequality Lemma A.1. Hence
∇u + ∇v p0
∇u − ∇v p0 1/(p−1) 1 1
k∇ukpp,a + k∇vkpp,a . (A.6)
+
≤ 2 2 2 2 p,a p,a 0
0
0
0
Similarly kukpp,b,∂Ω = k|u|p kp−1,b,∂Ω and kvkpp,b,∂Ω = k|v|p kp−1,b,∂Ω , so that, proceeding exactly as before and, applying again the reverse Minkowski inequality proved in Proposition A.5 of [4], we obtain
u + v p0
u − v p0 1/(p−1) 1 1
kukpp,b,∂Ω + kvkpp,b,∂Ω . (A.7) + ≤
2 2 2 2 p,b,∂Ω p,b,∂Ω Therefore, combining (A.6) with (A.7), we get
∇u + ∇v p0
∇u − ∇v p0
u + v p0
u − v p0
+
+
+
2 2 2 2 p,a p,a p,b,∂Ω p,b,∂Ω 1/(p−1) 1/(p−1) 1 1 1 1 ≤ k∇ukpp,a + k∇vkpp,a + kukpp,b,∂Ω + kvkpp,b,∂Ω 2 2 2 2 1 1/(p−1) 1 1 1 ≤ k∇ukpp,a + k∇vkpp,a + kukpp,b,∂Ω + kvkpp,b,∂Ω 2 2 2 2 1 1/(p−1) 1 = kuk + kvk , 2 2 since 1/(p − 1) > 1; in other words,
∇u + ∇v p0
u + v p0
(A.8)
+
2 2 p,a p,b,∂Ω
46
Giuseppina Autuori, Patrizia Pucci and Csaba Varga
u − v p0 1/(p−1) n ∇u − ∇v p0 o 1
kuk + kvk − .
+
2 2 2 2 p,a p,b,∂Ω Now, since ku − vk ≥ ε, it follows that 1/(p−1) 0 εp ≤ k∇u − ∇vkpp,a + ku − vkpp,b,∂Ω
u − v p0 ∇u − ∇v p0
, ≤ 22/(p−1)
+
2 2 p,a p,b,∂Ω since again 1/(p − 1) > 1. Hence,
u − v p0 ε p0 ∇u − ∇v p0
≤ + .
2/p 2 2 p,a p,b,∂Ω 2 ≤
1
0
Thus, choosing δ = δ(ε) such that 1 − ε/22/p = 2(p−2)/(p−1) (1 − δ)p , from (A.8), we obtain
∇u + ∇v p0
u + v p0 0
≤ 2(p−2)/(p−1) (1 − δ)p .
+
2 2 p,a p,b,∂Ω In conclusion, since again 1/(p − 1) > 1,
u + v p0
u + v p0 ∇u + ∇v p0 0
≤ (1 − δ)p ;
≤ 2(2−p)/(p−1)
+
2 2 2 p,a p,b,∂Ω that is, ku + vk ≤ 2(1 − δ), as required. Acknowledgments. This work was started while C. Varga was visiting the Universit` a degli Studi of Perugia in September 2009 and continued in September 2011 with GNAMPA–INdAM visiting professor positions. The first two authors have been partially supported by the MIUR project Metodi Variazionali ed Equazioni Differenziali alle Derivate Parziali Non Lineari. The third author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE2011-3-0241. References [1] R.A. Adams, “Sobolev Spaces,” Pure and Applied Mathematics, Vol. 65, Academic Press, New York–London, 1975. [2] W. Allegretto and Y.X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819–830. [3] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. [4] G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, to appear in NoDEA Nonlinear Differential Equations Appl. (2012), 34 pp. DOI 10.1007/s00030-012-0193-y
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