MULTIPLE SOLUTIONS OF GRADIENT-TYPE SYSTEMS ON THE SIERPINSKI GASKET BRIGITTE E. BRECKNER AND CSABA VARGA Abstract. Using a four-critical-point theorem obtained by B. Ricceri, we prove the existence of three nonzero weak solutions of certain gradient-type systems defined on the Sierpinski gasket.

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B. E. BRECKNER AND CS. VARGA

explicitly (e.g., in the proof of Theorem 2 in [10] this function, denoted by u2 , is actually a constant function, and in the proof of Theorem 1.1 in [7] this function, denoted by uc , is constructed with the aid of the distance function from a point to a certain compact subset of the open, bounded, and connected subset of Rn where the elliptic problem is considered), in our application of Ricceri’s theorem to the fractal case we have to involve Urysohn’s Lemma to prove the existence of such a function. Nevertheless, one can give concrete examples that allow an explicit construction of such functions, using a specific method that characterizes the Sierpinski gasket (see 1) in Example 4.7). Understanding and dealing with the phenomena in the case of gradient-type systems on the Sierpinski gasket is the first step for the study of these systems in the more general setting of post-critically-finite fractals. 2. Preliminaries Notations. 1) We denote by N the set of natural numbers {0, 1, 2, . . . }, by N∗ := N \ {0} the set of positive naturals, and by | · | the Euclidian norm on the spaces Rn , n ∈ N∗ . Throughout the paper, the spaces Rn are considered to be endowed with the Euclidean topology. 2) If X is a set, Y a nonempty subset of X, and f : X → R a real-valued function, then we will use the notations inf f := inf{f (y) | y ∈ Y } and sup f := sup{f (y) | y ∈ Y }. Y

Y ∗

3) If X is a real normed space, then X stands for its dual space. Our study of gradient-type systems on the Sierpinski gasket is based on the following abstract four-critical-point theorem by B. Ricceri which is a consequence of a more general result (see Theorem 1 in [10]). Theorem 2.1. Let X be a reflexive real Banach space, let Φ, Ψ, J : X → R be functionals, let z0 , z1 ∈ X, and let ρ > 0 be a real number such that the following conditions hold: (i) Φ is a coercive, sequentially weakly lower semicontinuous C 1 -functional whose derivative admits a continuous inverse on X ∗ . (ii) Ψ and J are C 1 -functionals with compact derivatives. (iii) z0 is a strict local minimum of the functional Φ and Φ(z0 ) = Ψ(z0 ) = J(z0 ) = 0. (iv) The inequalities ) ( J(u) J(u) , lim sup ≤0 (2.1) max lim sup u→z0 Φ(u) ||u||→∞ Φ(u) and (

(2.2)

Ψ(u) Ψ(u) max lim sup , lim sup u→z0 Φ(u) ||u||→∞ Φ(u)

) <1

3

hold. (v) 0 < J(z1 ) = sup(Φ−Ψ)−1 ( ]−∞,ρ]) J < supX J and Φ(z1 ) ≤ Ψ(z1 ). Then there exists λ∗ > 0 such that the functional Φ − λ∗ J − Ψ has at least four critical points, z0 being one of them. Moreover, two of these four critical points (different from z0 ) are actually global minima of Φ − λ∗ J − Ψ. Proof. Pick z2 ∈ X such that J(z1 ) < J(z2 ). According to (vi), the element z2 belongs to (Φ − Ψ)−1 ( ]ρ, ∞[ ). So, taking into account that ρ > 0, we get, by (v), that Φ(z1 ) − Ψ(z1 ) < ρ < Φ(z2 ) − Ψ(z3 ). Moreover, sup(Φ−Ψ)−1 ( ]−∞,ρ]) J − J(z2 ) sup(Φ−Ψ)−1 ( ]−∞,ρ]) J − J(z1 ) =0< . ρ − Φ(z1 ) + Ψ(z1 ) ρ − Φ(z2 ) + Ψ(z2 ) Using Remark 1 in [10], we get the following inequality sup inf (λ(Φ(x) − Ψ(x) − ρ) − J(x)) < inf sup(λ(Φ(x) − Ψ(x) − ρ) − J(x)). λ>0 x∈X

x∈X λ>0

An application of Theorem 1 in [10] finishes now the proof.



We briefly recall some notations which will be used in the sequel, and refer to sections 2–4 in [2] and in [3] for a more detailed presentation of these aspects. Throughout the paper, the letter V stands for the the Sierpinski gasket (SG for short) in RN −1 , where N ≥ 2 is a fixed natural number. There are two different approaches that lead to V , starting from given points p1 , . . . , pN ∈ RN −1 with |pi − pj | = 1 for i 6= j, and from the similarities Si : RN −1 → RN −1 , defined by 1 1 (2.3) Si (x) = x + pi , 2 2 for i ∈ {1, . . . , N }. While in the first approach the set V appears as the unique nonempty compact subset of RN −1 satisfying the equality (2.4)

V =

N [

Si (V ),

i=1

in the second one V is obtained as the closure of the set V∗ := V0 := {p1 , . . . , pN } and Vm :=

N [

S

m∈N

Vm , where

Si (Vm−1 ), for m ∈ N∗ .

i=1

In what follows V is considered to be endowed with the relative topology induced from the topology on RN −1 . The natural measure µ associated with V is the normalized restriction of the lnlnN2 -dimensional Hausdorff measure on RN −1 to the subsets of V . Thus µ(V ) = 1. The Lebesgue spaces Lp (V, µ), with p ≥ 1, are equipped with the usual || · ||p norm. The analog, in the case of the SG, of the usual Sobolev spaces is the real Hilbert space 1 H0 (V ), equipped with the inner product W : H01 (V ) × H01 (V ) → R which induces the

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B. E. BRECKNER AND CS. VARGA

norm || · || (see Section 3 in [2]). The space H01 (V ) can be compactly embedded in a space of continuous functions. More exactly, if one denotes by C(V ) the space of real-valued continuous functions on V , by C0 (V ) := {u ∈ C(V ) : u|V0 = 0}, and consider both spaces being endowed with the usual supremum norm || · ||sup , then the following Sobolev-type inequality holds for H01 (V ) (2.5)

||u||sup ≤ c||u||, for every u ∈ H01 (V ),

where c := 2N + 3. Moreover, the embedding (2.6)

(H01 (V ), || · ||) ,→ (C0 (V ), || · ||sup )

is compact. As in [2] and [3], ∆ : D → L2 (V, µ) denotes the weak Laplacian on V , where D is a certain linear subset of H01 (V ) which is dense in L2 (V, µ) (and dense also in (H01 (V ), ||·||)). Thus Z ∆u · vdµ, for every (u, v) ∈ D × H01 (V ).

−W(u, v) = V

We recall that ∆ is bijective, linear and self-adjoint. 3. Some basic facts about derivatives We start this section with a few straightforward results established in a general setting. They will be essential for determining the energy functional of gradient-type systems defined on the Sierpinski gasket. In what follows the term differentiable means Fr´echet differentiable and derivative means Fr´echet derivative. Let (X, || · ||X ) and (Y, || · ||Y ) be normed spaces over R such that Y ⊆ X and such that the inclusion (3.1)

i

(Y, || · ||Y ) ,→ (X, || · ||X )

is continuous. Define i∗ : X ∗ → Y ∗ by i∗ (x∗ ) = x∗ ◦ i. The continuity of i implies that of i∗ . The following result is now an immediate consequence of the chain rule. Proposition 3.1. Let y ∈ Y and assume that the map L : X → R is differentiable at y. Then the restriction ` : Y → R of L to Y is differentiable at y and `0 (y) = i∗ (L0 (y)). Corollary 3.2. Let L : X → R be differentiable on X. Then the restriction ` : Y → R of L to Y is differentiable and its derivative is given by `0 = i∗ ◦ L0 ◦ i. Corollary 3.3. Let L : X → R be a C 1 -functional. Then the restriction ` : Y → R of L to Y is also C 1 . Proof. We know from Corollary 3.2 that `0 = i∗ ◦ L0 ◦ i. The continuity of i∗ , L0 and i imply that of `0 . Hence ` is C 1 . 

5

Corollary 3.4. Assume that the inclusion from (3.1) is compact. If L : X → R is a C 1 -functional, then the restriction ` : Y → R of L to Y is a C 1 -functional with compact derivative. Proof. That ` is C 1 follows from Corollary 3.3. The compactness of `0 is a consequence of Corollary 3.2, since the composition of a continuous map (i∗ ◦ L0 in our case) with a compact one (i in our case) is compact.  Before applying these results in a concrete case, we make the following general assumption: Throughout the paper, if (X1 , || · ||1 ) and (X2 , || · ||2 ) are real normed spaces, then the product space X1 × X2 is assumed to be endowed with the norm || · || defined by p ||(x1 , x2 )|| = (||x1 ||1 )2 + (||x2 ||2 )2 . We mention also a notation that will be used in the sequel: If T : R2 → R admits partial derivatives with respect to both variables, then we agree to denote the partial derivative of T with respect to the first and the second variable with Ts and Tt , respectively. In order to apply the above results, retain the notations from the previous section and let X := C0 (V ) × C0 (V ) and Y := H01 (V ) × H01 (V ), where C0 (V ) is considered to be endowed with the norm || · ||sup , and H01 (V ) with the norm || · ||. We omit the straightforward proof of the following result (actually based on the mean value theorem and on the uniform continuity of continuous real-valued functions defined on compact sets). Proposition 3.5. Let γ ∈ L1 (V, µ), let T : R2 → R be a C 1 -function, and define the map L : C0 (V ) × C0 (V ) → R by Z γ(x)T (u(x)v(x))dµ. (3.2) L(u, v) = V 1

Then L is a C -functional and its differential, at a point (u0 , v0 ) ∈ C0 (V ) × C0 (V ), is given by Z Z 0 L (u0 , v0 )(h1 , h2 ) = γ(x)Ts (u0 (x), v0 (x))h1 (x)dµ + γ(x)Tt (u0 (x), v0 (x))h2 (x)dµ V

V

for all (h1 , h2 ) ∈ C0 (V ) × C0 (V ). Corollary 3.6. Let γ ∈ L1 (V, µ), let T : R2 → R be a C 1 -function, and define the map ` : H01 (V ) × H01 (V ) → R by Z `(u, v) = γ(x)T (u(x)v(x))dµ. V 1

Then ` is a C -functional with compact derivative and its differential, at a point (u0 , v0 ) ∈ H01 (V ) × H01 (V ), is given by Z Z 0 γ(x)Ts (u0 (x), v0 (x))h1 (x)dµ + γ(x)Tt (u0 (x), v0 (x))h2 (x)dµ ` (u0 , v0 )(h1 , h2 ) = V

V

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B. E. BRECKNER AND CS. VARGA

for all (h1 , h2 ) ∈ H01 (V ) × H01 (V ). Proof. Considering the functional L : C0 (V ) × C0 (V ) → R defined in (3.2), the map ` is the restriction of L to H01 (V ) × H01 (V ). From (2.5) and (2.6) we know that the embedding H01 (V ) × H01 (V ) ,→ C0 (V ) × C0 (V ) is continuous and compact. The assertion follows now from Proposition 3.5, Corollary 3.2 and Corollary 3.4.  4. One-parameter gradient-type systems on the SG We retain the notations from the previous sections. We start by considering the following functions: Let a, b ∈ L1 (V, µ) and let F, G : R2 → R be C 1 -functions. We formulate the following gradient-type system defined on the SG and depending on a positive real parameter λ: Find pairs of functions (u, v) with u, v : V → R satisfying certain properties and such that   −∆u(x) = λa(x)Fs (u(x), v(x)) + b(x)Gs (u(x), v(x)), ∀x ∈ V \ V0 ,     (Pλ ) −∆v(x) = λa(x)Ft (u(x), v(x)) + b(x)Gt (u(x), v(x)), , ∀x ∈ V \ V0 ,      u| = 0, v| = 0. V0 V0 We are interested in weak solutions of problem (Pλ ), i.e., in pairs (u, v) ∈ H01 (V ) × H01 (V ) satisfying the following system  R  W(u, h1 ) = V (λaFs (u, v) + bGs (u, v))h1 dµ, ∀ h1 ∈ H01 (V ), (4.1) R  W(v, h2 ) = V (λaFt (u, v) + bGt (u, v))h2 dµ, ∀ h2 ∈ H01 (V ). In order to apply variational methods for the study of problem (Pλ ) we have to introduce an energy functional. First of all we consider (according to the assumption made in the previous section) that the space H01 (V ) × H01 (V ) is endowed with the norm induced by the inner product (4.2)

((u, v), (e u, ve)) 7→ W(u, u e) + W(v, ve).

We denote this norm with the same symbol || · || as the norm on the space H01 (V ), since it will be clear from the context which of them is meant. In particular, H01 (V ) × H01 (V ) becomes this way a real Hilbert space. Define now Φ : H01 (V ) × H01 (V ) → R by (4.3)

1 Φ(u, v) = (||u||2 + ||v||2 ), 2

i.e., Φ(u, v) = 21 ||(u, v)||2 . A simple computation yields that Φ is differentiable and that its differential, at an arbitrary point (u0 , v0 ) ∈ H01 (V ) × H01 (V ), is given by (4.4)

Φ0 (u0 , v0 )(h1 , h2 ) = W(u0 , h1 ) + W(v0 , h2 ), ∀ (h1 , h2 ) ∈ H01 (V ) × H01 (V ).

7

Define next J, Ψ : H01 (V ) × H01 (V ) → R by Z a(x)F (u(x)v(x))dµ (4.5) J(u, v) = V

and Z (4.6)

Ψ(u, v) =

b(x)G(u(x)v(x))dµ, V

respectively. We know from Corollary 3.6 that J and Ψ are C 1 -functionals with compact derivatives. The next result is fundamental for the study of problem (Pλ ) via variational methods. Proposition 4.1. Let a, b ∈ L1 (V, µ), let F, G : R2 → R be C 1 -functions, and consider the maps Φ, J and Ψ defined in (4.3), (4.5) and (4.6), respectively. Then the following assertions are equivalent for (u0 , v0 ) ∈ H01 (V ) × H01 (V ): (i) (u0 , v0 ) is a critical point of Φ − λJ − Ψ. (ii) (u0 , v0 ) is a weak solution of problem (Pλ ). Proof. (i) =⇒ (ii) By assumption, (Φ − λJ − Ψ)0 (u0 , v0 )(h1 , h2 ) = 0, ∀ (h1 , h2 ) ∈ H01 (V ) × H01 (V ), so (Φ − λJ − Ψ)0 (u0 , v0 )(h1 , 0) = 0, ∀ h1 ∈ H01 (V ), and (Φ − λJ − Ψ)0 (u0 , v0 )(0, h2 ) = 0, ∀ h2 ∈ H01 (V ). Using Corollary 3.6 and formula (4.4), we get that (u0 , v0 ) is a weak solution of problem (Pλ ). (ii) =⇒ (i) If (u0 , v0 ) is a weak solution of (Pλ ), then it follows from (4.1), (4.4) and Corollary 3.6 that (u0 , v0 ) is a critical point of Φ − λJ − Ψ.  Remark 4.2. Corollary 4.1 shows that Φ − λJ − Ψ is an energy functional of (Pλ ), i.e., (u0 , v0 ) is a weak solution of problem (Pλ ) if and only if (u0 , v0 ) is a critical point of Φ − λJ − Ψ. Furthermore assume that K is a compact subset of V with K ∩ V0 = ∅ and µ(K) > 0. The next result emphasizes that one can associate with K a function uK ∈ H01 (V ) with certain properties. Proposition 4.3. There exists uK ∈ H01 (V ) such that uK (x) = 1, for every x ∈ K, and ||uK ||sup = 1. Proof. By Urysohn’s Lemma there exists a continuous function φ : V → [0, 1] such that φ(x) = 0 for x ∈ V0 and φ(x) = 1 for x ∈ K. According to Theorem 1.4.4 in [11] there

8

B. E. BRECKNER AND CS. VARGA

exists a function u e ∈ H01 (V ) with ||φ − u e||sup < 1. In particular, u e(x) 6= 0 for all x ∈ K. 1 Hence |e u(x)| > 0 for every x ∈ K. Note that |e u| ∈ H0 (V ), by Lemma 3.1 in [2]. Let ξ := min |e u(x)|. x∈K

Then ξ > 0. Define h : R → R by h(t) = min{t, ξ}. Since h is a Lipschitz mapping with h(0) = 0, Lemma 3.1 in [2] yields that h ◦ |e u| ∈ H01 (V ). We have that (h ◦ |e u|)(x) = ξ for 1 every x ∈ K. Thus uK := ξ (h ◦ |e u|) satisfies the required conditions.  Remark 4.4. We will see later that if K is a cell of level m of V (with m ∈ N, m ≥ 2), then one can construct in a natural way an element uK ∈ H01 (V ) with the properties required in Proposition 4.3. We recall how these cells emerge in the case of the SG: For m ∈ N∗ let Wm := ({1, . . . , N })m , and for w = (w1 , . . . , wm ) ∈ Wm put Sw := Sw1 ◦ · · · ◦ Swm , where the similarities Si , i ∈ {1, . . . , N }, are defined by (2.3). The equality (2.4) yields [ Sw (V ), V = w∈Wm

called the level m decomposition of V . Each Sw (V ), w ∈ Wm , is a cell of level m of V . The next result will be used in the proof of Theorem 4.6. Lemma 4.5. Let γ ∈ L1 (V, µ) be so that γ ≥ 0 a.e. on V , and let T : R2 → R be continuous with T (0, 0) = 0. Then the map ` : H01 (V ) × H01 (V ) → R, defined by Z γ(x)T (u(x)v(x))dµ, `(u, v) = V

satisfies the following inequalities (4.7)

( ) `(u, v) T (s, t) lim sup ≤ c2 ||γ||1 max 0, lim sup 2 2 + ||v||2 2 ||u|| (u,v)→(0,0) (s,t)→(0,0) s + t

and (4.8)

( ) `(u, v) T (s, t) lim sup ≤ c2 ||γ||1 max 0, lim sup 2 , 2 + ||v||2 2 ||u|| ||(u,v)||→∞ |(s,t)|→∞ s + t

where c is the constant in (2.5). Proof. Let η be an arbitrary real number so that ( ) T (s, t) (4.9) max 0, lim sup 2 < η. 2 (s,t)→(0,0) s + t Then there exists ρη > 0 such that (4.10)

T (s, t) ≤ η(s2 + t2 ), for all (s, t) ∈ R2 with |(s, t)| ≤ ρη .

Consider now (u, v) ∈ H01 (V ) × H01 (V ) with ||(u, v)|| ≤ ρcη . Using (2.5), one obtains that |(u(x), v(x))| ≤ ρη , for every x ∈ V . Involving also (4.10), we get T (u(x), v(x)) ≤ η(u2 (x) + v 2 (x)) ≤ η(||u||2sup + ||v||2sup ) ≤ ηc2 (||u||2 + ||v||2 ).

9

Since γ ≥ 0 a.e. on V , it follows that `(u, v) ≤ ηc2 ||γ||1 (||u||2 + ||v||2 ), hence

`(u, v) ≤ ηc2 ||γ||1 . 2 2 (u,v)→(0,0) ||u|| + ||v|| lim sup

Since the above inequality holds for every η satisfying (4.9), we finally obtain (4.7). In order to prove (4.8), let now η be an arbitrary real number so that ) ( T (s, t) < η. (4.11) max 0, lim sup 2 2 |(s,t)|→∞ s + t Then there exists δη > 0 such that T (s, t) ≤ η(s2 + t2 ), for all (s, t) ∈ R2 with |(s, t)| > δη . Let mη := max{T (s, t) : (s, t) ∈ R2 , |(s, t)| ≤ δη }. Then mη ≥ 0, hence T (s, t) ≤ η(s2 + t2 ) + mη , for all (s, t) ∈ R2 . If (u, v) ∈ H01 (V )×H01 (V ), then by the above inequality, combined with (2.5), one obtains, for every x ∈ V , that T (u(x), v(x)) ≤ η(u2 (x) + v 2 (x)) + mη ≤ ηc2 (||u||2 + ||v||2 ) + mη . Since γ ≥ 0 a.e. on V , it follows that `(u, v) ≤ ηc2 ||γ||1 (||u||2 + ||v||2 ) + mη ||γ||1 , hence

`(u, v) ≤ ηc2 ||γ||1 . 2 + ||v||2 ||u|| ||(u,v)||→∞ lim sup

Since the above inequality holds for every η satisfying (4.11), we finally obtain (4.8).  The following application of Theorem 2.1 was inspired by Theorem 2 in [10]. We had to adapt the methods involved in [10] in the context of the Sobolev space H 1 (0, 1) to the case of the function space H01 (V ). There are several differences between these two approaches. The major difficulty consists in the fact that (by contrast with the Sobolev space H 1 (0, 1)) nonzero constant functions don’t belong to H01 (V ). We managed this difficulty by introducing a certain compact subset K of V , and a certain function uK ∈ H01 (V ) associated with K. Theorem 4.6. Let a, b ∈ L1 (V, µ) \ {0}, let F, G : R2 → R be C 1 -functions, let K be a compact subset of V with K ∩ V0 = ∅ and µ(K) > 0, and let uK ∈ H01 (V ) be such that uK (x) = 1, for every x ∈ K, and ||uK ||sup = 1. Moreover, assume that the following conditions hold: (C1) a ≥ 0 and b ≥ 0 a.e. on V , (C2) a = 0 a.e. on V \ u−1 K (1).

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B. E. BRECKNER AND CS. VARGA

(C3) b = 0 a.e. on (supp uK ) \ K. (C4) F (0, 0) = G(0, 0) = 0. (C5) F satisfies the inequalities F (s, t) F (s, t) ≤ 0 and lim sup 2 ≤ 0. 2 2 2 (s,t)→(0,0) s + t |(s,t)|→∞ s + t lim sup

(C6) G satisfies the inequalities G(s, t) 1 and M := sup G(s, t) < ∞. < 2 2 2c2 ||b||1 (s,t)→(0,0) s + t R2 p (C7) There exist a real number σ > c 2M ||b||1 and a pair (ξ1 , ξ2 ) ∈ R2 with |(ξ1 , ξ2 )| ≤ σ such that lim sup

0 < F (ξ1 , ξ2 ) = max{F (s, t) : (s, t) ∈ R2 , |(s, t)| ≤ σ} < sup F

(4.12)

R2

and (4.13)

1 ||uK ||2 |(ξ1 , ξ2 )|2 ≤ G(ξ1 , ξ2 ) 2

Z bdµ. K

Then there exists λ∗ > 0 such that problem (Pλ∗ ) has at least three nonzero weak solutions. Moreover, two of these three solutions are global minima of the energy functional of problem (Pλ∗ ) Proof. Set X := H01 (V ) × H01 (V ), endowed (as mentioned above) with the norm || · || induced from the inner product defined in (4.2). Consider the functions Φ, Ψ, J : X → R defined in (4.3), (4.5) and (4.6), respectively. Denote by u0 ∈ H01 (V ) the constant zero function. Put u1 := ξ1 uK and v1 := ξ2 uK . Let z0 := (u0 , u0 ) ∈ X, z1 := (u1 , v1 ) ∈ X and σ2 ρ := 2c 2 − M ||b||1 . Assumption (C7) implies that ρ > 0. We show next that conditions (i)–(v) of Theorem 2.1 are fulfilled: The conditions (i) and (iii) are obviously satisfied, while (ii) follows from Corollary 3.6. The assumption (C5) yields, according to Lemma 4.5, inequality (2.1). The first inequality from (C6) implies, together with Lemma 4.5, that Ψ(u, v) lim sup < 1. (u,v)→z0 Φ(u, v) From the second inequality in (C6) we get that G(s, t) ≤ 0, 2 2 |(s,t)|→∞ s + t lim sup

which yields, again by an application of Lemma 4.5, that Ψ(u, v) ≤ 0. ||(u,v)||→∞ Φ(u, v) lim sup

So inequality (2.2) is also fulfilled. Thus condition (iv) of Theorem 2.1 is satisfied, too.

11

We are going to verify condition (v) of Theorem 2.1: For this fix an arbitrary pair (u, v) ∈ (Φ − Ψ)−1 ( ] − ∞, ρ]). Since b ≥ 0 a.e. on V , we obtain from the second inequality in (C6) that Ψ(u, v) ≤ M ||b||1 , so Φ(u, v) − M ||b||1 ≤ ρ, and, by the definition of Φ, ||u||2 + ||v||2 ≤ 2(ρ + M ||b||1 ). Involving now (2.5), we obtain that ||u||2sup + ||v||2sup ≤ 2c2 (ρ + M ||b||1 ) = σ 2 . Our arguments from above show that the inclusion (4.14)

(Φ − Ψ)−1 ( ] − ∞, ρ]) ⊆ {(u, v) ∈ X : ||u||2sup + ||v||2sup ≤ σ 2 }

holds true. We next observe that by (C2) Z Z aF (ξ1 uK , ξ2 uK )dµ = J(z1 ) =

u−1 K (1)

V

aF (ξ1 uK , ξ2 uK )dµ = F (ξ1 , ξ2 )||a||1 .

Since a 6= 0, inequality (4.12) yields in particular that (4.15)

J(z1 ) > 0.

Pick now an arbitrary pair (u, v) ∈ X such that ||u||2sup +||v||2sup ≤ σ 2 . Then |(u(x), v(x))| ≤ σ, for every x ∈ V . Using (4.12), we thus get F (u(x), v(x)) ≤ F (ξ1 , ξ2 ), ∀ x ∈ V. Since a ≥ 0 a.e. on V , it follows that J(u, v) ≤ F (ξ1 , ξ2 )||a||1 = J(z1 ). Hence we get, using (4.14), that (4.16)

J(u, v) ≤ J(z1 ), ∀ (u, v) ∈ (Φ − Ψ)−1 ( ] − ∞, ρ]).

Furthermore, by (C3), (C4) and (4.13), we obtain that 1 Φ(z1 ) − Ψ(z1 ) = ||uK ||2 |(ξ1 , ξ2 )|2 − G(ξ1 , ξ2 ) 2 hence (4.17)

Φ(z1 ) − Ψ(z1 ) ≤ 0 < ρ.

Moreover, in view of (4.16) and (4.17), we conclude that (4.18)

J(z1 ) =

sup (Φ−Ψ)−1 ( ]−∞,ρ])

J.

Z bdµ ≤ 0, K

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B. E. BRECKNER AND CS. VARGA

According to (4.12), there is a pair (ξ10 , ξ20 ) ∈ R2 such that F (ξ1 , ξ2 ) < F (ξ10 , ξ20 ). Put z2 := (ξ10 uk , ξ20 uK ) ∈ X. Using (C2), we obtain that J(z2 ) = F (ξ10 , ξ20 )||a||1 . Thus J(z1 ) < J(z2 ), showing that (4.19)

J(z1 ) < sup J. X

The relations (4.15), (4.17), (4.18), and (4.19) finally imply that condition (v) of Theorem 2.1 is also satisfied. An application of this theorem finishes the proof.  Examples 4.7. Before presenting concrete situations that allow an application of Theorem 4.6, we have to recall the definition of the quadratic form W , that corresponds to the inner product W : H01 (V ) × H01 (V ) → R. For this one considers first the quadratic forms Wm , m ∈ N, defined for functions um : Vm → R, m ∈ N, by  m X N +2 (4.20) Wm (um ) = (u(x) − u(y))2 . N x,y∈V m |x−y|=2−m

Then W (u) = lim Wm (u|Vm ), hence ||u|| = m→∞

p

W (u), for u ∈ H01 (V ).

1) Subsets K ⊆ V with the properties required in Theorem 4.6 occur, for instance, as cells in the level m decomposition of V , for m ∈ N with m ≥ 2 (see Remark 4.4): For such an m let w = (w1 , . . . , wm ) ∈ Wm be so that there exist i, j ∈ {1, . . . , m} with wi 6= wj . It can be easily proved that Sw (V ) ∩ V0 = ∅. Put K := Sw (V ). The scaling property 2.1 in [6] yields that µ(K) > 0. The map uK ∈ H01 (V ) with uK (x) = 1, for all x ∈ K, and ||uK ||sup = 1 can be chosen, for instance, to be the map obtained by successive harmonic extensions (see Section 1.3 in [11]), followed by a continuous continuation, from the map u eK : Vm → R defined by  0, x ∈ Vm \ Sw (V0 ) u eK (x) = 1, x ∈ Sw (V0 ). The successive harmonic extensions give rise to a (unique) map u∗k : V∗ → R. This map admits a unique continuation uK ∈ H01 (V ) (see Section 1.4 in [11]). Moreover, 0 ≤ uK (x) ≤ p 1, for all x ∈ V , uK (x) = 1, for all x ∈ Sw (V ) = K, and ||uK || = Wm (e uK ), where Wm (e uK ) is defined in (4.20). We illustrate these aspects for N = 3. In this case V0 consists of the three points in Fig. 1 which are the vertices of an equilateral triangle with |p1 − p2 | = 1. Consider now, for instance, m = 2 and w = (1, 2). Then K = (S1 ◦ S2 )(V ). Fig. 2 illustrates the values of the map u eK at the points of V2 . It also indicates the triangle in which K is included. We obtain in this case that  2 5 50 2 ||uK || = W2 (e uK ) = (2 + 2 + 2) = . 3 3

13

2) Suppose now that a compact set K ⊆ V and a function uK ∈ H01 (V ) with the properties mentioned in Theorem 4.6 have been fixed, i.e., K ∩ V0 = ∅, µ(K) > 0, uK (x) = 1, for every x ∈ K, and ||uK ||sup = 1. We illustrate a possibility for choosing the functions a, b, F, and G satisfying the conditions required in Theorem 4.6. For this we proceed in the following steps: 1. Pick a ∈ L1 (V, µ) \ {0} with a ≥ 0 a.e. on V and a = 0 a.e. on V \ K. 2. Fix (ξ1 , ξ2 ) ∈ R2 \ {(0, 0)}. 3. Let G : R2 → R be a C 1 -function such that G(s, t) G(0, 0) = 0, G(ξ1 , ξ2 ) > 0, lim sup 2 ≤ 0, and M := sup G(s, t) < ∞. 2 (s,t)→(0,0) s + t R2 4. Fix σ > 0 so that c2 ||uK ||2

(4.21)

M σ2 < . G(ξ1 , ξ2 ) |(ξ1 , ξ2 )|2

Put α := c2 ||uK ||2 G(ξM1 ,ξ2 ) . Since ||uK ||sup = 1, we get from (2.5) that c2 ||uK || ≥ 1. Thus α ≥ 1, which yields, in particular, that |(ξ1 , ξ2 )| < σ. 5. Let F : R2 → R be a C 1 -function such that F (s, t) F (s, t) F (0, 0) = 0, lim sup 2 ≤ 0, lim sup 2 ≤ 0, 2 2 (s,t)→(0,0) s + t |(s,t)|→∞ s + t and 0 < F (ξ1 , ξ2 ) = max{F (s, t) : (s, t) ∈ R2 , |(s, t)| ≤ σ} < sup F. R2

14

B. E. BRECKNER AND CS. VARGA

Such a function can be constructed starting, for instance, from a C 1 -function f : R → R with the following properties • f (0) = 0 and 0 is a local maximum point of f , • f (x) ≤ f (ξ12 + ξ22 ), for every x ∈ [0, σ 2 ], and 0 < f (ξ12 + ξ22 ), • f (ξ12 + ξ22 ) < sup f , [0,∞[

• there exists d > 0 such that f (x) ≤ 0, for every x ≥ d. Define now F : R2 → R by F (s, t) = f (s2 + t2 ). Then F is a C 1 function satisfying the above conditions. 6. Finally choose b ∈ L1 (V, µ) \ {0} such that b ≥ 0 a.e. on V , b = 0 a.e. on V \ K, and ||uK ||2 (ξ12 + ξ22 ) σ2 ≤ ||b||1 < 2 . 2G(ξ1 , ξ2 ) 2c M Note that this inequality can be fulfilled according to (4.21). The functions a, b, F, and G satisfy the assumptions of Theorem 4.6. References [1] G. Bonanno, G. Molica Bisci, and V. R˘ adulescu, Qualitative analysis of gradient-type systems with oscillatory nonlinearities on the Sierpinski gasket, Chin. Ann. Math. 34 B (3) (2013), 381–398. [2] B. E. Breckner, V. R˘ adulescu, and Cs. Varga, Infinitely many solutions for the Dirichlet problem on the Sierpinski gasket, Anal. Appl. (Singap.) 9 (2011), 235–248. [3] B. E. Breckner, D. Repovˇs, and Cs. Varga, On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket, Nonlinear Anal. 73 (2010), 2980-2990. [4] D. G. de Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, in: Handbook of Differential Equations: Stationary Partial Differential Equations, volume 5 (edited by M. Chipot), Elsevier, 2008, 1–48. [5] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd edition, John Wiley & Sons, 2003. [6] F. Faraci, A. Krist´ aly, Three non-zero solutions for a nonlinear eigenvalue problem, J. Math. Anal. Appl. 394 (1) (2012), 225–230. [7] A. Krist´ aly, V. R˘ adulescu, and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Cambridge University Press, Cambridge, 2010. [8] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401–410. [9] B. Ricceri, A class of nonlinear eigenvalue problems with four solutions, J. Nonlinear Convex Anal. 11 (2010), 503–511. [10] R.S. Strichartz, Differential Equations on Fractals. A Tutorial, Princeton University Press, Princeton, NJ, 2006. ˘ lniceanu Babes¸-Bolyai University, Faculty of Mathematics and Computer Science, Koga str. 1, 400084 Cluj-Napoca, Romania E-mail address: [email protected] ˘ lniceanu Babes¸-Bolyai University, Faculty of Mathematics and Computer Science, Koga str. 1, 400084 Cluj-Napoca, Romania E-mail address: varga gy [email protected]

## MULTIPLE SOLUTIONS OF GRADIENT-TYPE ...

NJ, 2006. Babes-Bolyai University, Faculty of Mathematics and Computer Science, KogËalniceanu str. 1, 400084 Cluj-Napoca, Romania. E-mail address: [email protected]gmx.net. Babes-Bolyai University, Faculty of Mathematics and Computer Science, KogËalniceanu str. 1, 400084 Cluj-Napoca, Romania. E-mail address: varga ...

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