On multiple solutions for multivalued elliptic equations under Navier boundary conditions C. O. Alves

J. Abrantes Santos

[email protected]

J. V. Goncalves

[email protected]

[email protected]

Dedicated to Professor Pl´ acido Zoega T´ aboas on the occasion of his 70th birthday

1.

Abstract

We employ variational methods for non-smooth functionals to show existence of multiple solutions for multivalued fourth order elliptic equations under Navier boundary conditions. Our main result extends similar ones known for the Laplacian.

There exist numbers t± ∈ R with t− < 0 < t+ such that Z
(t±)2 (f4) . F (x, t±φ1) − F∞(x) > µ1 2 Ω Then (1.1) admits at least three non-trivial solutions, say u−, u+, u0 ∈ H satisfying   α∆2u + β∆u ∈ [f (x, u(x)), f (x, u(x))] a.e. x ∈ Ω, 

2.

Introduction

We deal with existence and multiplicity of solutions of the problem

(1.1)

∆2u =

N X i,j=1

b Φ(u) = Q(u) − Ψ(u), u ∈ H.

I(u0) = inf max I(γ(t)) > 0,

Since Q is a C 1−functional, we derive

where h0, vi = hQ0(u0), viH − hµ0, viH , v ∈ H,

Γ = {γ ∈ C([0, 1], H) | γ(0) = 0, γ(1) = t+φ1}. for some

∂ 4u . 2 2 ∂ xi ∂ xj 3.

The boundary condition Bu = 0 on ∂Ω means that

b 0). µ0 ∈ ∂ Ψ(u

Abstract Framework That is

u = 0 on ∂Ω if α = 0 and u = ∆u = 0 on ∂Ω if α > 0, (trace sense). Let X be a real Banach space. A functional I : X → R is locally Lipschitz continuous, I ∈ Liploc(X, R) for short, if given u ∈ X there is an open neighborhood V := Vu ⊂ X and some constant K = KV > 0 such that | I(v2) − I(v1) |≤ K k v2 − v1 k, vi ∈ V, i = 1, 2.

For each function u ∈ L2(Ω), we set Z s Z F (x, s) = f (x, t)dt and Ψ(u) = F (x, u)dx, 0

Ω

where f : Ω × R → R is a suitable measurable function. The functional Ψ(u) is locally Lipschitz continuous and its subdifferential is denoted by ∂Ψ(u).

The directional derivative of I at u in the direction of v ∈ X is defined by I(u + h + λv) − I(u + h) 0 I (u; v) = lim sup . λ h→0, λ↓0

By a solution of (1.1) we mean an element u ∈ H := H01(Ω) ∩ H 2(Ω) such that

Our aim is to find multiple solutions of (1.1), under the condition

Ω

and corresponding norm

As a consequence of the inequality below Z Z | ∆u |2≥ λ1 | ∇u |2,

(1.3)

Ω

H is a Hilbert space, details in Section 4. It will be shown that the solutions of (1.1) are the critical points (in a suitable sense) of the energy functional Z Z Z   1 F (x, u), u ∈ H. | ∆u |2 −β | ∇u |2 − I(u) = α 2 Ω Ω Ω In order to establish our main result we set f (x, t) = lim inf f (x, s), f (x, t) = lim sup f (x, s) s→t

and

(f2) (ii) Our main result is

µ1 2
|t|→∞ F (x, t) − t −→ F∞(x) a.e. x ∈ Ω, 2 b | F (x, t) |≤ µ21 t2 + H(x) a.e. x ∈ Ω.

Theorem 1 Assume that f : Ω × R → R is measurable, f (x, 0) = 0 a.e. x ∈ Ω and satisfies (f1)(i)(ii), (f2)(i)(ii). Assume, in addition, the conditions: There exist m ∈ L∞(Ω), δ > 0, 0 ≤ m < µ1, m 6≡ 0 such that, (f3) (i) (ii)

 0 ∂I(u) = I (u) , if I ∈ C 1(X, R).

µ2 2 F (x, t) ≤ t , a.e. x ∈ Ω, t ∈ R, where µ2 = λ2(αλ2 − β), 2 m(x)t2 F (x, t) ≤ , a.e. x ∈ Ω, |t| ≤ δ. 2

we assume that α > 0, (the other case is standard). Since Z Z Z α ∆u0∆v − β ∇u0∇v = ξv, v ∈ H, Ω

Ω

Ω

we have, from the Generalized Green Identity ∆u0 = 0 on ∂Ω, (trace sense). Let I ∈ Liploc(X, R) and assume that C ⊂X is convex c is a real number. The non-smooth functional I satisfies the (P S)c,C condition if any sequence (un) ⊂ C such that n→∞

n→∞

I(un) −→ c and m(un) −→ 0, admits a subsequence which converges to some point of C.

References

Assume in addition that

Then I admits a local minimum u ∈ int(C) if

[1] D. G. Costa and J. V. Goncalves, Critical Point theory for Nondifferentiable Functionals and Applications, J. Math. Anal. Appl. 153 (1990) 470-485.

I satisfies (P S)c,C .

[2] F.H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 265 (1975), 247-262.

I(e u) < inf I(u) for some u e ∈ int(C). ∂C

(3.1)

4. Liploc Functionals and Results on Multivalued Equations

max{| f (x, t) − µ1t |, | f (x, t) − µ1t |} −→ 0 a.e. x ∈ Ω,

(i)

Bu0 = 0 on ∂Ω in sense of trace,

u∈C

s→t

| f (x, t) − µ1t |≤ τ (x) a.e. x ∈ Ω,

So there is ξ ∈ L2(Ω) such that µ0 = ξ. By the elliptic regularity theory u0 ∈ H 4(Ω) and

c = inf I(u).

|t|→∞

(ii)

Ω

µ0v, v ∈ H.

µ0 ∈ [f (x, u0(x)), f (x, u0(x))] a.e. x ∈ Ω.

k µ kX ∗ ; µ ∈ ∂I(u) ,

The theorem below improves results by Mizoguchi [8], Goncalves & Miyagaki [5]. Theorem 2 Let I : X → R be locally Lipschitz continuous, bounded from below. Assume that X is reflexive and C ⊂ X is a convex, closed set such that int(C) 6= ∅. Set

and we shall assume that there are functions τ b ∈ L1(Ω) satisfying the following basic conL1(Ω) with F∞ ≥ 0 and H ditions:

(f1)

Ω

∇u0∇v −

By Theorem 3,



A critical point of I is an element u0 ∈ X such that 0 ∈ ∂I(u0) and a critical value of I is a real number c such that I(u0) = c for some critical point u0 ∈ X. If an element u0 ∈ X is a local minimum of I ∈ Liploc(X; R) then it is a critical point of I. For each u, v ∈ X,

∈ L2(Ω), F∞ ∈

(i)

Ω

∆u0∆v − β

In order to show that

I 0(u; v) = max{hµ, vi | µ ∈ ∂I(u)}.

kuk2 = hu, uiH .

Ω

m(u) = min

0=α

Z

α∆2u0(x) + β∆u0(x) ∈ [f (x, u0(x)), f (x, u0(x))] a.e. x ∈ Ω.

∂I(u) ⊂ X ∗ is convex, non-empty and weak*-compact, 

Z

Therefore,

A few definitions and properties will be recalled below.

f (x, t) = µ1 , lim t |t|→+∞ where µ1 := λ1(αλ1 − β) is the first eigenvalue of the eigenvalue problem   α∆2u + β∆u = µu in Ω, (1.2)  Bu = 0 on ∂Ω, In order to establish our main result we need some notations and definitions which. At first consider the space H endowed with the inner product Z Z hu, viH = α ∆u∆v − β ∇u∇v, u, v ∈ H

Z

α∆2u0 + β∆u0 = ξ a.e. in Ω.

The generalized gradient of I at u is the set  ∗ 0 ∂I(u) = µ ∈ X ; hµ, vi ≤ I (u; v), v ∈ X .

α∆2u + β∆u ∈ ∂Ψ(u) and Bu = 0 on ∂Ω.

Ω

1 Q(u) = k u k2, u ∈ H. 2

Then

γ∈Γ 0≤t≤1

1(Ω))), where α ≥ 0, −∞ < β < αλ1, (λk is the k th eigenvalue of (−∆, H R 20 and the principal λ1-eigenfunction is φ1, normalized such that Ω φ1dx = 1,

Bu0 = 0 on ∂Ω, (trace sense).

Proof. Let

and

Bu = 0 on ∂Ω,

If u0 ∈ H is a critical point of Φ, then u0 ∈ H 4(Ω) and   α∆2u0(x) + β∆u0(x) ∈ [f (x, u0(x)), f (x, u0(x))] a.e. x ∈ Ω 

Ω

  α∆2u + β∆u ∈ ∂Ψ(u) in Ω, 

Bu = 0 on ∂Ω (trace sense), Z I(u+) = min{I(v) | v ∈ H, vφ1 > 0} < 0, ZΩ I(u−) = min{I(v) | v ∈ H, vφ1 < 0} < 0,

Proposition 4 Assume (f1)(ii) and (f5) and set Z 1 Φ(u) = k u k2 − F (x, u), u ∈ H. 2 Ω

The result below will be used in the sequence and the reader is referred to Chang [9], Costa and Goncalves [1] for further details. Theorem 3 Assume that f : Ω×R → R is measurable satisfying (f1)(ii) and f , f : Ω × R → R are N-measurable, that is, for each u ∈ L2(Ω), we have (f5)

x 7→ f (x, u(x)) and x 7→ f (x, u(x)) are Lebesgue measurable.

If

Z Ψ(u) =

F (x, u), u ∈ L2(Ω)

Ω

then Ψ : L2(Ω) → R is Liploc with ∂Ψ(u) ⊂ [f (x, u(x)), f (x, u(x))], a.e. x ∈ Ω. b ≡ Ψ |H we have Moreover, setting Ψ b ∂ Ψ(u) ⊂ ∂Ψ(u), u ∈ H.

[3] J. V. Goncalves and O. Miyagaki, Existence of nontrivial solutions for semilinear elliptic equations at resonance, Hauston J. Math. 16 (1990) 583-594. [4] J. V. Goncalves and O. Miyagaki, Multiple nontrivial solutions of semilinear strongly resonant elliptic equations, Nonlinear Anal. 19 (1992) 43-52. [5] J. V. Gon¸calves and O. Miyagaki, Three solutions for a strongly resonant elliptic problem, Nonlinear Anal. 24 (1995) 265-272. [6] C. P. Gupta and Y. G. Kwong, Biharmonic eigenvalue problems and Lp estimates, J. Math. Math. Sci. 13 (1990) 469-480. [7] F. H. Clarke, Optimization and nonsmooth analysis, SIAM, Philadelphia, 1990. [8] N. Mizoguchi, Existence of nontrivial solutions of partial differential equations with discontinuous nonlinearities, Nonlinear Anal. 16 (1991) 1025-1034. [9] K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. App. 80 (1981) 102-129. [10] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349-381.