Existence Results for the p (cid:2) x (cid:3) -Laplacian with Nonlinear Boundary Condition

By using the variational method, under appropriate assumptions on the perturbation terms f (cid:2) x,u (cid:3) ,g (cid:2) x,u (cid:3) such that the associated functional satisﬁes the global minimizer condition and the fountain theorem, respectively, the existence and multiple results for the p (cid:2) x (cid:3) -Laplacian with nonlinear boundary condition in bounded domain Ω were studied. The discussion is based on variable exponent Lebesgue and Sobolev spaces.


Introduction
In recent years, increasing attention has been paid to the study of differential and partial differential equations involving variable exponent conditions. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, or calculus of variations. For more information on modeling physical phenomena by equations involving p x -growth condition we refer to 1-3 . The appearance of such physical models was facilitated by the development of variable exponent Lebesgue and Sobolev spaces, L p x and W 1, p x , where p x is a real-valued function. Variable exponent Lebesgue spaces appeared for the first time in the literature as early as 1931 in an article by Orlicz 4 . The spaces L p x are special cases of Orlicz spaces L ϕ originated by Nakano 5 and developed by Musielak and Orlicz 6,7 , where f ∈ L ϕ if and only if ϕ x, |f x | dx < ∞ for a suitable ϕ. Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers. In that context we refer to the studies of Tsenov 8 , Sharapudinov 9 , and Zhikov 10, 11 .

ISRN Applied Mathematics
In this paper, we consider the following nonlinear elliptic boundary value problem: − div a x |∇u| p x −2 ∇u b x |u| p x −2 u λf x, u , x ∈ Ω, a x |∇u| p x −2 ∂u ∂ν c x |u| q x −2 u μg x, u , x ∈ ∂Ω, where Ω ⊂ R n is a bounded domain with Lipschitz boundary ∂Ω, ∂/∂ν is outer unit normal derivative, p x ∈ C Ω , q x ∈ C ∂Ω , p x , q x > 1, and p x / q y for any x ∈ Ω, y ∈ ∂Ω; λ, μ ∈ R; f : Ω × R → R, and g : ∂Ω × R → R are Carathédory functions. Throughout this paper, we assume that a x , b x , and c x satisfy The operator −Δ p x u : − div |∇u| p x −2 ∇u is called p x -Laplacian, which is a natural extension of the p-Laplace operator, with p being a positive constant. However, such generalizations are not trivial since the p x -Laplace operator possesses a more complicated structure than the p-Laplace operator, for example, it is inhomogeneous. For related results involving the Laplace operator, see 12, 13 . In the past decade, many people have studied the nonlinear boundary value problems

1.2
Bonder and Rossi 14 considered the existence of nontrivial solutions of problem 1.2 when f x, u ≡ 0 and discussed different cases when g x, u is subcritical, critical, and supercritical with respect to u. We also mention that Martínez and Rossi 15 studied the existence of solutions when p q and the perturbation terms f x, u and g x, u satisfy the Landesman-Lazer-type conditions. Recently, J.-H. Zhao and P.-H. Zhao 16 studied the nonlinear boundary value problem, assumed that f x, u and g x, u satisfy the Ambrosetti-Rabinowitz-type condition, and got the multiple results.
If λ μ 1, p x ≡ p, and q x ≡ q a constant , then problem 1.1 becomes

1.3
There are also many people who studied the p-Laplacian nonlinear boundary value problems involving 1.3 . For example, Cîrstea and Rǎdulescu 17 used the weighted Sobolev space to discuss the existence and nonexistence results and assumed that f x, u is a special case in the problem 1.3 , where Ω is an unbounded domain. Pflüger 18 , by using the same technique, considered the existence and multiplicity of solutions when b x ≡ 0. The author showed the existence result when f x, u and g x, u are superlinear and satisfy the Ambrosetti-Rabinowitz-type condition and got the multiplicity of solutions when one of f x, u and g x, u is sublinear and the other one is superlinear.
More recently, the study on the nonlinear boundary value problems with variable exponent has received considerable attention. For example, Deng 19 studied the eigenvalue of p x -Laplacian Steklov problem, and discussed the properties of the eigenvalue sequence under different conditions. Fan 20 discussed the boundary trace embedding theorems for variable exponent Sobolev spaces and some applications. Yao 21 constrained the two nonlinear perturbation terms f x, u and g x, u in appropriate conditions and got a number of results for the existence and multiplicity of solutions. Motivated by Yao and problem 1.3 , we consider the more general form of the variable exponent boundary value problem 1.1 . Under appropriate assumptions on the perturbation terms f x, u and g x, u , by using the global minimizer method and fountain theorem, respectively, the existence and multiplicity of solutions of 1.1 were obtained. These results extend some of the results in 21 and the classical results for the p-Laplacian in 14, 16, 22-24 .

Preliminaries
In order to discuss problem 1.1 , we need some results for the spaces W 1, p x Ω , which we call variable exponent Sobolev spaces. We state some basic properties of the spaces W 1, p x Ω , which will be used later for more details, see 25, 26 . Let Ω be a bounded domain of R n , and denote We can also denote C ∂Ω and q , q − for any q x ∈ C ∂Ω , and define with norms on L p x Ω and L p x ∂Ω defined by where dσ is the surface measure on ∂Ω. Then, L p x Ω , | · | p x and L p x ∂Ω , | · | L p x ∂Ω become Banach spaces, which we call variable exponent Lebesgue spaces. Let us define the space then, from the assumptions of a x and b x , it is easy to check that u is an equivalent Hence, we have see 27 where ξ 1 , ξ 2 and ζ 1 , ζ 2 are positive constants independent of u. If q x ∈ C Ω and q x < p * x for any x ∈ Ω, then the embedding from W 1, p x Ω into L q x Ω is compact and continuous, where

2.10
3 If q x ∈ C ∂Ω and q x < p * x for any x ∈ ∂Ω, then the trace imbedding from W 1, p x Ω into L q x ∂Ω is compact and continuous, where

Assumptions and Statement of Main Results
In the following, let X denote the generalized Sobolev space W 1,p x Ω , X * denote the dual space of W 1,p x Ω , · denote the dual pair, and let → represent strong convergence, represent weak convergence, C, C i represent the generic positive constants. Now we state the assumptions on perturbation terms f x, u and g x, u for problem 1.1 as follows: f 0 f : Ω × R → R satisfies Carathéodory condition and there exist two constants where α x ∈ C Ω and α x < p * x for any x ∈ Ω.
g 0 g : ∂Ω × R → R satisfies Carathéodory condition and there exist two constants where β x ∈ C ∂Ω and β x < p * x for any x ∈ ∂Ω.
The functional associated with problem 1.1 is where F x, u and G x, u are denoted by By Propositions 3.1 and 3.2, and assumptions f 0 , g 0 , it is easy to see that the functional ϕ ∈ C 1 X, R ; moreover, ϕ is even if f 2 and g 3 hold. Then, so the weak solution of 1.1 corresponds to the critical point of the functional ϕ. Before giving our main results, we first give several propositions that will be used later.

Proposition 3.1 see 31 . If one denotes
then I ∈ C 1 X, R and the derivative operator of I, denoted by I , is and one has: i I : X → X * is a continuous, bounded, and strictly monotone operator, ii I is a mapping of (S+) type, that is, if u n u in X and lim sup n → ∞ I u n − I u , u n − u ≤ 0, then u n → u in X, iii I : X → X * is a homeomorphism.

Proposition 3.2 see 19 . If one denotes
where q x ∈ C ∂Ω and q x < p * x for any x ∈ ∂Ω, then J ∈ C 1 X, R and the derivative operator J of J is and one has that J : X → R and J : X → X * are sequentially weakly-strongly continuous, namely, u n u in X implies J u n → J u .

ISRN Applied Mathematics
Let X be a reflexive and separable Banach space. There exist e i ∈ X and e * j ∈ X * such that X span{e i : i 1, 2, . . .}, X * span e * j : j 1, 2, . . . , One important aspect of applying the standard methods of variational theory is to show that the functional ϕ satisfies the Palais-Smale condition, which is introduced by the following definition. In what follows we write the PS c condition simply as the PS condition if it holds for every level c ∈ R for the Palais-Smale condition at level c. Proposition 3.4 Fountain theorem, see 23,32 . Assume that A1 X is a Banach space, ϕ ∈ C 1 X, R is an even functional, the subspaces X k , Y k and Z k are defined by 3.13 .Suppose that, for every k ∈ N, there exist ρ k > γ k > 0 such that A4 ϕ satisfies PS c condition for every c > 0.
Then, ϕ has a sequence of critical values tending to ∞.
Proof. By Propositions 2.2 and 2.3, we know that if we denote then Φ is weakly continuous and its derivative operator, denoted by Φ , is compact. By Propositions 3.1 and 3.2, we deduce that ϕ I − J − Φ is also of S type. To verify that ϕ satisfies PS condition on X, it is enough to verify that any PS sequence is bounded. Suppose that {u n } ⊂ X such that ϕ u n −→ c, ϕ u n −→ 0, in X * , as n −→ ∞. 3.17 Then, for n large enough, we can find M 3 > 0 such that

3.18
Since ϕ u n → 0, we have ϕ u n , u n → 0. In particular, { ϕ u n , u n } is bounded. Thus, there exists M 4 > 0 such that ϕ u n , u n ≤ M 4 .

3.19
We claim that the sequence {u n } is bounded. If it is not true, by passing a subsequence if necessary, we may assume that u n → ∞. Without loss of generality, we assume that u n ≥ 1 appropriately large such that ξ 1 u p − < ζ 1 u p for any x ∈ Ω. From 3.18 and 3.19 and letting θ min{θ 1 , θ 2 }, then θ < q − , we have

ISRN Applied Mathematics
By virtue of assumptions f 1 and g 1 and combining 3.20 and 3.21 , we have

3.22
Note that θ min{θ 1 , θ 2 } > p , let n → ∞ we obtian a contradiction. It follows that the sequence {u n } is bounded in X. Therefore, ϕ satisfies PS condition.
Under appropriate assumptions on the perturbation terms f x, u , g x, u , a sequence of weak solutions with energy values tending to ∞ was obtained. The main result of the paper reads as follows.
Proof. We will prove that ϕ satisfies the conditions of Proposition 3.4. Obviously, because of the assumptions of f 2 and g 2 , ϕ is an even functional and satisfies PS condition see Lemma 3.6 . We will prove that if k is large enough, then there exist ρ k > γ k > 0 such that A2 and A3 hold. By virtue of f 0 , g 0 , there exist two positive constants C 1 , C 2 such that

3.23
Letting u ∈ Z k with u > 1 appropriately large such that ξ 1 u p − < ζ 1 u p , we have

11
If max{|u| q L q x ∂Ω , |u| L q x ∂Ω , then by Proposition 3.5 , we have
By virtue of f 1 and g 1 , there exist two positive constants C 4 , C 5 such that

3.28
Letting u ∈ Y k , we have

3.29
If max{ξ 2 u p , Since dim Y k < ∞, all norms are equivalent in Y k . So we get Also, note that q − > θ 1 , θ 2 > p , Then, we get ϕ u → −∞ as u → ∞. For other cases, the proofs are similar and we omit them here. So A3 holds. From the proof of A2 and A3 , we can choose ρ k > γ k > 0. Thus, we complete the proof.
This time our idea is to show that ϕ possesses a nontrivial global minimum point in X.
Proof. Firstly, we show that ϕ is coercive. For sufficiently large norm of u u ≥ 1 , and by virtue of 3.23 ,

3.34
So ϕ is coercive since α , β , q < p − . Secondly, by Proposition 2.2, it is easy to verify that ϕ is weakly lower semicontinuous. Thus, ϕ is bounded below and ϕ attains its infimum in X, that is, ϕ u 0 inf u∈X ϕ u and u 0 is a critical point of ϕ, which is a weak solution of 1.1 .
In the Theorem 3.8, we cannot guarantee that u 0 is nontrivial. In fact, under the assumptions on the above theorem, we can also get a nontrivial weak solution of ϕ.