Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition

1 Introduction

The double-phase functional of the calculus of variations

was introduced by Zhikov in [25] in connection with homogenization, strongly anisotropic materials, elasticity, Lavrentiev phenomenon, and so on. Following [25], for strongly anisotropic materials, the regulation of the mixture between two different materials, with p and q hardening, is modulated by the coefficient a = a ( x ) . Independently, even without the presence of the modulating coefficient a = a ( x ) , for the regularity of minimizers, Marcellini in [15] proves that a condition of the type q p < 1 + o ( n ) is sufficient, while Giaquinta in [14] establishes the necessary part.

The articles [3,4] present sharp regularity theorems for minimizers of different double-phase integral models (1.1). In [7], the minimizers are proved to be C 1 , β -regular under appropriate conditions and further regularity results for related minimizers are given in [8] and in the more recent paper [10] and in the references therein.

Concerning existence results involving double-phase operators in bounded domains, we refer to [2,16], and in the whole space to [12], as well as to the references cited there.

In recent years, variable exponential problems have become a research hotpot because of its wide applications, for instance, in image restoration, image denoising, and image enhancement of image processing, etc. More application background can be found in [6,13,19–21]. Variable exponential double-phase nonlinear problems describe objects, which contain two different materials involving power hardening exponents p ( x ) and q ( x ) . Energy functionals of this kind of problems are double-phase functionals, either isotropic or anisotropic. Indeed, these problems have been widely studied, and we refer to [5,18,22,23] and the references therein. In [22], Shi et al. deal with the equation:

with λ > 0 and μ ≥ 0 . They use the weighting method to overcome the lack of compactness and prove that the aforementioned equation has two pairs of nontrivial nonnegative and nonpositive solutions as well as infinitely many nontrivial solutions. Since they prove that the corresponding energy functional satisfies the Palais-Smale condition, in short PS, the function g ( x , u ) is required to satisfy the A-R condition. The previous article [23] deals with the following equation:

where the differential operator − div A ( x , ∇ u ) satisfies condition (1.2), which means the p ( ⋅ ) -material is present if ∣ ξ ∣ > 1 and the q ( ⋅ ) -material acts if ∣ ξ ∣ ≤ 1 . In [23], the weighted variable exponent spaces associated with p ( x ) and q ( x ) are introduced to prove two existence theorems of weak solutions without the A-R condition. In fact, this article improves [23], because there is one more term h ( x ) ∣ u ∣ r ( x ) − 2 u in the equation ( ℰ ) than in the equation studied in [23]. Clearly, the addition of this term requires to use a new solution function space. Of course, although the proof ideas are somehow similar, the subsequent analysis and calculations are also different.

From now on, for v : R N → R , we define

and write v 1 ≪ v 2 to represent

Motivated by [22] and [23], in this article, we study the equation

where A : R N × R N → R N admits a potential A , with respect to its second variable ξ , satisfying the following assumptions.

( A 1 ) the potential A = A ( x , ξ ) is a continuous function in R N × R N , with continuous derivative with respect to ξ , A = ∂ ξ A ( x , ξ ) , and verifies:

1. A ( x , 0 ) = 0 and A ( x , ξ ) = A ( x , − ξ ) , for all ( x , ξ ) ∈ R N × R N ;

2. A ( x , ⋅ ) is strictly convex in R N for all x ∈ R N ;

3. there exist positive constants C 1 , C 2 and variable exponents p and q such that for all ( x , ξ ) ∈ R N × R N

   (1.2) C 1 ∣ ξ ∣ p ( x ) , if ∣ ξ ∣ > 1 C 1 ∣ ξ ∣ q ( x ) , if ∣ ξ ∣ ≤ 1 ≤ A ( x , ξ ) ⋅ ξ ,

   and

   (1.3) ∣ A ( x , ξ ) ∣ ≤ C 2 ∣ ξ ∣ p ( x ) − 1 , if ∣ ξ ∣ > 1 C 2 ∣ ξ ∣ q ( x ) − 1 , if ∣ ξ ∣ ≤ 1 ;

4. 1 ≪ p ≪ q ≪ min { N , p ∗ } in R N , and p , q are Lipschitz continuous functions in R N ;

5. A ( x , ξ ) ⋅ ξ ≤ s ( x ) A ( x , ξ ) for any ( x , ξ ) ∈ R N × R N , where s is a Lipschitz continuous function in R N , with q ≤ s ≪ p ∗ in R N ;

( A 2 ) A is uniformly convex, that is, for any ε ∈ ( 0 , 1 ) , there exists δ ( ε ) ∈ ( 0 , 1 ) such that either ∣ u − v ∣ ≤ ε max { ∣ u ∣ , ∣ v ∣ } or A x , u + v 2 ≤ 1 2 ( 1 − δ ( ε ) ) ( A ( x , u ) + A ( x , v ) ) for any x , u , v ∈ R N .

When A ( x , ∇ u ) = ∣ ∇ u ∣ p ( x ) − 2 ∇ u , if ∣ ∇ u ∣ > 1 ∣ ∇ u ∣ q ( x ) − 2 ∇ u , if ∣ ∇ u ∣ ≤ 1 , then

is an operator of p ( x ) -Laplacian type and the corresponding potential is

Lemma A.2 in [23], see also Lemma 2.4.7 of [9], guarantees that the potential A satisfies ( A 1 ) and ( A 2 ), provided that 1 ≪ p ≪ q ≪ N in R N .

In addition, we also require the validity of

( ℋ V )

1. V ∈ L loc 1 ( R N ) ;

2. V ≥ V 0 > 0 a.e in R N ;

3. V ( x ) → ∞ as ∣ x ∣ → ∞ .

( ℋ h )

1. h ∈ L loc 1 ( R N ) ;

2. h ≥ h 0 > 0 a.e. in R N ;

3. h ( x ) → ∞ as ∣ x ∣ → ∞ .

( ℋ f 1 ) α , r ∈ C ( R N ) , with α − , r − > 1 , and α + , r + < ∞ , 0 ≤ f ( x , t ) t , f ( x , t ) t = o ( ∣ t ∣ α ( x ) ) and f ( x , t ) t = o ( ∣ t ∣ r ( x ) ) as t → 0 , and ∣ f ( x , t ) ∣ ≤ C ( 1 + ∣ t ∣ γ ( x ) ) , where γ is a Lipschitz continuous function, with max { α , r } ≪ γ ≪ p ∗ .

( ℋ f 2 ) There exist positive constants M , C 1 , C 2 , and a function a ≫ q in R N such that for all t , with ∣ t ∣ ≥ M , and for all x ∈ R N

where F ( x , t ) = ∫ 0 t f ( x , τ ) d τ .

The application of the mountain pass lemma to obtain a critical point of a functional φ , which is a nontrivial solution of the underlying equation, requires that the nonlinear term f satisfies the A-R condition [1], which ensures that the energy functional φ satisfies the PS condition, i.e., any sequence ( u n ) n such that φ ( u n ) → c and φ ′ ( u n ) → 0 has a convergent subsequence. In this article, we use the Cerami condition (see Lemma 3.8) to replace the PS condition, so that f only needs to satisfy conditions ( ℋ f 1 )–( ℋ f 2 ), which are weaker than the A-R condition. For example, f ( x , t ) = ∣ t ∣ s ( x ) − 2 t [ ln ( 1 + ∣ t ∣ ) ] a ( x ) , with s ≥ q , a ≫ q in R N , satisfies assumptions ( ℋ f 1 )–( ℋ f 2 ), but it does not verify the A-R condition.

Equation ( ℰ ) has been studied in [17], where α ( x ) = p ( x ) , f ( x , u ) = λ w ( x ) ∣ u ∣ q ( x ) − 2 u and where − div A ( x , ∇ u ) is only associated with the p ( x ) -Laplacian. More precisely, [17] deals with existence of entire solutions for it under suitable assumptions. In the present article, we study existence of a pair of nontrivial nonnegative and nonpositive solutions of ( ℰ ). For this purpose, we need to introduce an appropriate Sobolev space that matches the double-phase structure of the elliptic operator and the weighted term V ( x ) ∣ u ∣ α ( x ) − 2 u . Meanwhile, we also need to consider the effects of h ( x ) ∣ u ∣ r ( x ) − 2 u in the space. This brings new difficulties to handle. In addition, there are many differences in calculation from the previous work. In the rest of this article, we first present the weighted variable exponent spaces, involving the weighted term h ( x ) and then the mountain pass lemma. Our main result can be stated as follows:

This article is divided into four sections. Section 2 presents some properties of function spaces with variable exponent; Section 3 presents some properties of functionals and operators; Section 4 presents the proof of Theorem 1.1.

2 Variable exponent space theory

Throughout the article, the letters c , c i , C , C i , i = 1 , 2 , … , denote positive constants, which may vary from line to line, but are independent of the terms which will take part in any limit process.

To discuss the equation ( ℰ ), we need some properties of the variable exponent Lebesgue and Sobolev spaces. Let Ω ⊂ R N be an open domain. Let S ( Ω ) be the set of all measurable real-valued functions defined on Ω . We write as follows:

where p ∈ C + ( Ω ¯ ) . The space L p ( ⋅ ) ( Ω ) is equipped with the norm

and ( L p ( ⋅ ) ( Ω ) , ∣ u ∣ L p ( ⋅ ) ( Ω ) ) is a Banach space, called generalized Lebesgue space. If Ω = R N , then we drop R N in our notations if there is no ambiguity. For example, we simply denote the space L p ( ⋅ ) ( R N ) and the norm on it as L p ( ⋅ ) and ∣ u ∣ L p ( ⋅ ) .

The variable exponent Sobolev space W 1 , p ( ⋅ ) ( Ω ) is defined by

and it is equipped with the norm

for all u ∈ W 1 , p ( ⋅ ) ( Ω ) . While W 0 1 , p ( ⋅ ) ( Ω ) is the closure of C 0 ∞ ( Ω ) in W 1 , p ( ⋅ ) ( Ω ) .

Since equation ( ℰ ) is a double-phase problem, we need to define the following variable exponent Sobolev space.

Throughout this article, we denote

Conditions ( A 1 )-(i) and (iii) imply that for all ( x , ξ ) ∈ R N × R N

This estimate, in combination with (1.3) and ( A 1 )-(i) and (ii), yields

In what follows, we discuss ( ℰ ) in the variable exponent spaces ( E , ‖ ⋅ ‖ E ) and ( X , ‖ ⋅ ‖ X ).

Let E = { u ∈ L V α ( ⋅ ) ∣ ∇ u ∈ ( L p ( ⋅ ) + L q ( ⋅ ) ) N } with the norm

where

is endowed with the norm

Let

be endowed with the norm

Let ψ ∈ C ∞ ( R N ) , with ψ ( x ) = 1 if x ∈ R N and ∣ x ∣ ≤ 1 , while ψ ( x ) = 0 if x ∈ R N and ∣ x ∣ ≥ 3 , and 0 ≤ ψ ≤ 1 and ∣ ∇ ψ ( x ) ∣ ≤ 1 in R N .

Let n ∈ N and ψ n ( x ) = ψ ( x ∕ n ) . Set A n = { x ∈ R N ∣ n ≤ ∣ x ∣ ≤ 3 n } , A n c = R N ⧹ A n , B n = { x ∈ R N ∣ ∣ x ∣ ≤ n } , and B n c = { x ∈ R N ∣ ∣ x ∣ > n } .

Let u ∈ L ∞ and define u n = ψ n u ; thus, u n ∈ L ∞ and ∣ u n ∣ L ∞ ≤ ∣ u ∣ L ∞ . Clearly, u n has compact support and u n ∈ L V α ( ⋅ ) . Since ∇ u n = ψ n ∇ u + u ∇ ψ n , then ∇ u n ∈ L p ( ⋅ ) + L q ( ⋅ ) if both terms of the sum are in L p ( ⋅ ) + L q ( ⋅ ) . Now, ψ n ∈ L ∞ and ∇ u ∈ L p ( ⋅ ) + L q ( ⋅ ) , so that ψ n ∇ u ∈ L p ( ⋅ ) + L q ( ⋅ ) . Since ∇ ψ n vanishes in A n c , ∣ A n ∣ < ∞ , ∇ ψ n ∈ ( L ∞ ) N and u ∈ L ∞ , then u ∇ ψ n ∈ L p ( ⋅ ) + L q ( ⋅ ) . In conclusion, u n ∈ X loc ∩ L ∞ .