Combined Effects of Concave-Convex Nonlinearities in a Fourth-Order Problem with Variable Exponent

1 Introduction

In a pioneering paper, A. Ambrosetti, H. Brezis and G. Cerami [1] initiated the qualitative analysis of semilinear Dirichlet elliptic problems that involve concave and convex nonlinearities. They proved several existence, multiplicity and nonexistence results and developed powerful topological and variational methods for the study of such nonlinear problems. In particular, they studied the effects of small perturbations for the existence of solutions. In [17, 13] related existence results are established in the case of elliptic problems with variable exponents and Dirichlet boundary condition (see [26, 28] for further developments and related properties). The main purpose of this paper is to complete the results of L. Kong [13] and to prove the existence of a family of eigenvalues in a neighborhood of the origin. We also refer to the related papers [10, 18, 27, 29, 30]. Additional results on higher-order problems or nonlinear partial differential equations with variable exponent can be found in the papers by G. Autuori, F. Colasuonno and P. Pucci [3], Z. Chen [5], F. Colasuonno and P. Pucci [6], A. Kratohvil and I. Necas [14], V. Lubyshev [16], P. Pucci and Q. Zhang [24].

Let Ω⊂ℝN be a bounded domain with smooth boundary. Consider the following nonhomogeneous eigenvalue problem:

where p,q:Ω¯→ℝ are continuous functions and Δp⁢(x) denotes the p⁢(x)-Laplace operator, which is defined by

Δ p ⁢ ( x ) ⁢ u := div ⁡ ( | ∇ ⁡ u | p ⁢ ( x ) - 2 ⁢ ∇ ⁡ u ) .

Problem (1.1) is studied in [17] (see also [28, Section 2.3.1]) in a subcritical setting under the basic assumption

1 < min x ∈ Ω ¯ ⁡ q ⁢ ( x ) < min x ∈ Ω ¯ ⁡ p ⁢ ( x ) < max x ∈ Ω ¯ ⁡ q ⁢ ( x ) .

Under this hypothesis, the main result in [17] establishes that there exists λ*>0 such that problem (1.1) has at least one nontrivial solution for all λ∈(0,λ*). Since the associated energy functional does not have a mountain pass geometry (see A. Ambrosetti and P. Rabinowitz [2]), the proof relies essentially on the Ekeland variational principle, see [9]. We point out that the original proof of the mountain pass theorem is based on several powerful deformation techniques developed by R. Palais and S. Smale [20, 21], who developed the main ideas of the Morse theory in the abstract framework of differential topology on infinite-dimensional Riemann manifolds. A simpler proof of the mountain pass theorem is due to H. Brezis and L. Nirenberg [4], who used a pseudo-gradient lemma, a perturbation argument and the Ekeland variational principle.

The study initiated in [17, 28] was continued by L. Kong [13] in the framework of the p⁢(x)-biharmonic operatorΔp⁢(x)2, namely

Δ p ⁢ ( x ) 2 ⁢ u := Δ ⁢ ( | Δ ⁢ u | p ⁢ ( x ) - 2 ⁢ Δ ⁢ u ) .

Consider the fourth-order nonlinear elliptic equation with variable exponent and Dirichlet boundary condition

where a⁢(x) and w⁢(x) are nonnegative potentials and the nonlinear term f behaves like

f ⁢ ( u ) = | u | γ ⁢ ( x ) - 2 ⁢ u - | u | β ⁢ ( x ) - 2 ⁢ u ,

where γ,β>1 are continuous functions, and we assume the basic hypothesis

The main result in [13] asserts that there exists λ*>0 such that problem (1.2) has at least one nontrivial solution for all λ∈(0,λ*).

In the present paper, we establish several existence results for problems related to (1.2) but under some basic assumptions different from (1.3).

We consider the nonlinear problem

where λ is a positive parameter and a≥0. Under two different assumptions, we show that problem (1.4) has at least one nontrivial solution if the positive parameter λ is small enough. The proof relies on the Ekeland variational principle and the mountain pass theorem. We refer to J. Garcia Azorero and I. Peral Alonso [11] who applied the mountain pass theorem to obtain the existence of a nodal (that is, sign-changing) solution in a related quasilinear setting.

The situation changes if we consider a problem very close to (1.4). Let us consider the following eigenvalue nonlinear Dirichlet problem:

In this case, we establish a sufficient condition for the existence of nontrivial solutions provided that the parameter λ is large enough. The proof is based on the direct method of the calculus of variations.

In Section 2 we recall some basic definitions and properties concerning the basic function spaces with variable exponent. We refer to the recent monographs of L. Diening, P. Hästo, P. Harjulehto and M. Ruzicka [8] and V. Rădulescu and D. Repovš [28] for related properties of Lebesgue and Sobolev spaces with variable exponents. The main results are stated in Section 3 of this paper. Final comments and some open problems are given in Section 4.

2 Function Spaces with Variable Exponent

Consider the set

C + ( Ω ¯ ) = { p ∈ C ( Ω ¯ ) ; p ( x ) > 1 for all x ∈ Ω ¯ } .

For all p∈C+⁢(Ω¯) we define

p + = sup x ∈ Ω ⁡ p ⁢ ( x )   and   p - = inf x ∈ Ω ⁡ p ⁢ ( x ) .

For any p∈C+⁢(Ω¯), we define the variable exponent Lebesgue space

L p ⁢ ( x ) ( Ω ) = { u ; u is measurable and ∫ Ω | u ( x ) | p ⁢ ( x ) d x < ∞ } .

This vector space is a Banach space if it is endowed with the Luxemburg norm, which is defined by

| u | p ⁢ ( x ) = inf { μ > 0 ; ∫ Ω | u ⁢ ( x ) μ | p ⁢ ( x ) d x ≤ 1 } .

The function space Lp⁢(x)⁢(Ω) is reflexive if and only if 1<p-≤p+<∞. Continuous functions with compact support are dense in Lp⁢(x)⁢(Ω) if p+<∞.

Let Lq⁢(x)⁢(Ω) denote the conjugate space of Lp⁢(x)⁢(Ω), where 1/p⁢(x)+1/q⁢(x)=1. If u∈Lp⁢(x)⁢(Ω) and v∈Lq⁢(x)⁢(Ω), then the following Hölder-type inequality holds:

| ∫ Ω u ⁢ v ⁢ 𝑑 x | ≤ ( 1 p - + 1 q - ) ⁢ | u | p ⁢ ( x ) ⁢ | v | q ⁢ ( x ) .

Moreover, if pj∈C+⁢(Ω¯) (j=1,2,3) and

1 p 1 ⁢ ( x ) + 1 p 2 ⁢ ( x ) + 1 p 3 ⁢ ( x ) = 1 ,

then, for all u∈Lp1⁢(x)⁢(Ω), v∈Lp2⁢(x)⁢(Ω), w∈Lp3⁢(x)⁢(Ω),

| ∫ Ω u ⁢ v ⁢ w ⁢ 𝑑 x | ≤ ( 1 p 1 - + 1 p 2 - + 1 p 3 - ) ⁢ | u | p 1 ⁢ ( x ) ⁢ | v | p 2 ⁢ ( x ) ⁢ | w | p 3 ⁢ ( x ) .

The inclusion between Lebesgue spaces also generalizes the classical framework, namely if 0<|Ω|<∞ and p1, p2 are variable exponents so that p1≤p2 in Ω, then there exists the continuous embedding

L p 2 ⁢ ( x ) ⁢ ( Ω ) ↪ L p 1 ⁢ ( x ) ⁢ ( Ω ) .

If k is a positive integer number and p∈C+⁢(Ω¯), we define the variable exponent Sobolev space by

W k , p ⁢ ( x ) ( Ω ) = { u ∈ L p ⁢ ( x ) ( Ω ) ; D α u ∈ L p ⁢ ( x ) ( Ω ) for all | α | ≤ k } .

Here α=(α1,…,αN) is a multi-index, |α|=∑i=1Nαi and

D α ⁢ u = ∂ | α | ⁡ u ∂ x 1 α 1 ⁡ … ⁢ ∂ x N α N .

On Wk,p⁢(x)⁢(Ω) we consider the norm

∥ u ∥ k , p ⁢ ( x ) = ∑ | α | ≤ k | D α ⁢ u | p ⁢ ( x ) .

Then Wk,p⁢(x)⁢(Ω) is a reflexive and separable Banach space. Let W0k,p⁢(x)⁢(Ω) denote the closure of C0∞⁢(Ω) in Wk,p⁢(x)⁢(Ω).

Consider the function space E defined by

E = W 0 1 , p ⁢ ( x ) ⁢ ( Ω ) ∩ W 2 , p ⁢ ( x ) ⁢ ( Ω ) .

Then E is a separable and reflexive Banach space if it is equipped with the norm

∥ u ∥ E = ∥ u ∥ 1 , p ⁢ ( x ) + ∥ u ∥ 2 , p ⁢ ( x ) .

The norms ∥u∥E and |Δ⁢u|p⁢(x) are equivalent (cf. [13, p. 251]).

If a is a positive number, define, for all u∈E,

∥ u ∥ a = inf { λ > 0 ; ∫ Ω ( | Δ ⁢ u λ | p ⁢ ( x ) + a | u λ | p ⁢ ( x ) ) d x ≤ 1 } .

Then ∥u∥a is well-defined and it is a norm which is equivalent with the norms ∥u∥E and |Δ⁢u|p⁢(x) in E.

Let ϱa:E→ℝ be the modular function defined by

ϱ a ⁢ ( u ) = ∫ Ω ( | Δ ⁢ u | p ⁢ ( x ) + a ⁢ | u | p ⁢ ( x ) ) ⁢ 𝑑 x .

If (un), u∈E, then the following properties are true:

Let p*⁢(x) denote the critical Sobolev exponent, namely

p * ⁢ ( x ) = { N ⁢ p ⁢ ( x ) N - 2 ⁢ p ⁢ ( x ) if 2 ⁢ p ⁢ ( x ) < N , + ∞ if 2 ⁢ p ⁢ ( x ) ≥ N .

We point out that if p,q∈C+⁢(Ω¯) and q⁢(x)<p⋆⁢(x) for all x∈Ω¯, then the embedding E↪Lq⁢(x)⁢(Ω) is compact, see [13, Proposition 1.3].

The variable exponent Lebesgue and Sobolev spaces are generalizations of the classical Lebesgue and Sobolev spaces, replacing the constant exponent p with an exponent function p⁢(⋅). These spaces have been the subject of constant interest since the beginning of the 20th century both as function spaces with intrinsic interest and for their applications to problems arising in nonlinear partial differential equations and the calculus of variations. We refer to the monographs [7, 8, 28] for related properties of these spaces and their history.

3 The Main Results and Related Properties

We say that λ is an eigenvalue of problem (1.4) if there exists u∈E∖{0} such that, for all v∈E,

∫ Ω | Δ ⁢ u | p ⁢ ( x ) - 2 ⁢ Δ ⁢ u ⁢ Δ ⁢ v ⁢ 𝑑 x + a ⁢ ∫ Ω | u | p ⁢ ( x ) - 2 ⁢ u ⁢ v ⁢ 𝑑 x = λ ⁢ ∫ Ω ( | u | γ ⁢ ( x ) - 2 ⁢ u ⁢ v - | u | β ⁢ ( x ) - 2 ⁢ u ⁢ v ) ⁢ 𝑑 x .

If λ is an eigenvalue of problem (1.4), the corresponding function u∈E∖{0} is a weak solution of problem (1.4).

We study problem (1.4) under one of the following hypotheses:

or

The energy functional associated to problem (1.4) is defined as

ℰ λ ⁢ ( u ) = ∫ Ω 1 p ⁢ ( x ) ⁢ ( | Δ ⁢ u | p ⁢ ( x ) + a ⁢ | u | p ⁢ ( x ) ) ⁢ 𝑑 x - λ ⁢ ∫ Ω [ | u | γ ⁢ ( x ) γ ⁢ ( x ) - | u | β ⁢ ( x ) β ⁢ ( x ) ] ⁢ 𝑑 x   for all ⁢ u ∈ E .

Hypothesis (3.1) implies that ℰλ is well-defined, of class C1, and

〈 ℰ λ ′ ⁢ ( u ) , v 〉 = ∫ Ω ( | Δ ⁢ u | p ⁢ ( x ) - 2 ⁢ Δ ⁢ u ⁢ Δ ⁢ v + a ⁢ | u | p ⁢ ( x ) - 2 ⁢ u ⁢ v ) ⁢ 𝑑 x - λ ⁢ ∫ Ω ( | u | γ ⁢ ( x ) - 2 - | u | β ⁢ ( x ) - 2 ) ⁢ u ⁢ v ⁢ 𝑑 x   for all ⁢ v ∈ E .

The first result of this paper is the following.

We are then concerned with the study of problem (1.5). We say that λ is an eigenvalue of problem (1.5) if there exists u∈E∖{0} such that

∫ Ω | Δ ⁢ u | p ⁢ ( x ) - 2 ⁢ Δ ⁢ u ⁢ Δ ⁢ v ⁢ 𝑑 x + a ⁢ ∫ Ω | u | p ⁢ ( x ) - 2 ⁢ u ⁢ v ⁢ 𝑑 x = λ ⁢ ∫ Ω | u | γ ⁢ ( x ) - 2 ⁢ u ⁢ v ⁢ 𝑑 x - ∫ Ω | u | β ⁢ ( x ) - 2 ⁢ u ⁢ v ⁢ 𝑑 x   for all ⁢ v ∈ E .

We point out that hypothesis (3.1) implies that problem (1.4) does not have a mountain pass geometry. More precisely, ℰλ satisfies one of the geometric hypotheses of the mountain pass theorem, namely the existence of a “mountain” between two prescribed “villages”. However, the second geometric assumption of the mountain pass theorem is not fulfilled because this “valley” is close to the first “village” and not across the chain of mountains, as requested by the mountain pass theorem. For this reason the existence of the solution follows with different arguments and only for small perturbations (in terms of λ). An interesting open problem is to provide a complete description for all values of the positive parameter λ.

We remark that Theorem 3.1 establishes a property related to [13, Theorem 2.1]. However, our result is based on the assumption (3.1), which is more general than the corresponding hypothesis (2.1) in [13].

The proofs of Theorems 3.1 and 3.2 use some ideas developed in [17, 28, 27] in the framework of p⁢(x)-Laplace operators and extended in [13] to biharmonic operators with variable exponent.