Existence Results for the p(x)-Laplacian with Nonlinear Boundary Condition
By using the variational method, under appropriate assumptions on the perturbation terms such that the associated functional satisfies the global minimizer condition and the fountain theorem, respectively, the existence and multiple results for the -Laplacian with nonlinear boundary condition in bounded domain Ω were studied. The discussion is based on variable exponent Lebesgue and Sobolev spaces.
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 -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, and , where 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 . The spaces are special cases of Orlicz spaces originated by Nakano  and developed by Musielak and Orlicz [6, 7], where if and only if for a suitable . 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 , Sharapudinov , and Zhikov [10, 11].
In this paper, we consider the following nonlinear elliptic boundary value problem: where is a bounded domain with Lipschitz boundary is outer unit normal derivative, , and for any , and are Carathédory functions. Throughout this paper, we assume that , and satisfy and .
The operator is called -Laplacian, which is a natural extension of the -Laplace operator, with being a positive constant. However, such generalizations are not trivial since the -Laplace operator possesses a more complicated structure than the -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 involving -Laplacian. For example, if and (a constant), then problem (1.1) becomes Bonder and Rossi  considered the existence of nontrivial solutions of problem (1.2) when and discussed different cases when is subcritical, critical, and supercritical with respect to . We also mention that Martínez and Rossi  studied the existence of solutions when and the perturbation terms and satisfy the Landesman-Lazer-type conditions. Recently, J.-H. Zhao and P.-H. Zhao  studied the nonlinear boundary value problem, assumed that and satisfy the Ambrosetti-Rabinowitz-type condition, and got the multiple results.
If and (a constant), then problem (1.1) becomes There are also many people who studied the -Laplacian nonlinear boundary value problems involving (1.3). For example, Cîrstea and Rǎdulescu  used the weighted Sobolev space to discuss the existence and nonexistence results and assumed that is a special case in the problem (1.3), where is an unbounded domain. Pflüger , by using the same technique, considered the existence and multiplicity of solutions when . The author showed the existence result when and are superlinear and satisfy the Ambrosetti-Rabinowitz-type condition and got the multiplicity of solutions when one of and 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  studied the eigenvalue of -Laplacian Steklov problem, and discussed the properties of the eigenvalue sequence under different conditions. Fan  discussed the boundary trace embedding theorems for variable exponent Sobolev spaces and some applications. Yao  constrained the two nonlinear perturbation terms and 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 and , 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  and the classical results for the -Laplacian in [14, 16, 22–24].
In order to discuss problem (1.1), we need some results for the spaces , which we call variable exponent Sobolev spaces. We state some basic properties of the spaces , which will be used later (for more details, see [25, 26]). Let be a bounded domain of , and denote For write We can also denote and for any , and define with norms on and defined by where is the surface measure on . Then, and become Banach spaces, which we call variable exponent Lebesgue spaces. Let us define the space equipped with the norm For , if we define then, from the assumptions of and , it is easy to check that is an equivalent norm on . For simplicity, we denote
Hence, we have (see )(i)if , then ,(ii)if , then ,where and are positive constants independent of .
Denote by the closure of in .
If , , for any , then and the imbedding is continuous.
Proposition 2.2 (see [20, 21, 28]). are separable reflexive Banach spaces.
If and for any , then the embedding from into is compact and continuous, where
If and for any , then the trace imbedding from into is compact and continuous, where
(Poincaré inequality) There is a constant , such that
Proposition 2.5 (see ). Denote Then, (1) implies ,(2) implies .
3. Assumptions and Statement of Main Results
In the following, let denote the generalized Sobolev space , denote the dual space of , denote the dual pair, and let represent strong convergence, represent weak convergence, , represent the generic positive constants.
Now we state the assumptions on perturbation terms and for problem (1.1) as follows: ) satisfies Carathéodory condition and there exist two constants such that where and for any .() There exist such that ().() satisfies Carathéodory condition and there exist two constants such that where and for any . There exist such that .
The functional associated with problem (1.1) is where and are denoted by By Propositions 3.1 and 3.2, and assumptions , , it is easy to see that the functional ; moreover, is even if and ) hold. Then, so the weak solution of (1.1) corresponds to the critical point of the functional .
Before giving our main results, we first give several propositions that will be used later.
Proposition 3.1 (see ). If one denotes then and the derivative operator of , denoted by , is and one has:(i) is a continuous, bounded, and strictly monotone operator,(ii) is a mapping of (S+) type, that is, if in and , then in ,(iii) is a homeomorphism.
Proposition 3.2 (see ). If one denotes where and for any , then and the derivative operator of is and one has that and are sequentially weakly-strongly continuous, namely, in implies .
Let be a reflexive and separable Banach space. There exist and such that For , denote
One important aspect of applying the standard methods of variational theory is to show that the functional satisfies the - condition, which is introduced by the following definition.
Definition 3.3. Let and . Then, functional satisfies the condition if any sequence such that
contains a subsequence converging to a critical point of .
In what follows we write the condition simply as the condition if it holds for every level for the - condition at level .
Proposition 3.4 (Fountain theorem, see [23, 32]). Assume that(A1) is a Banach space, is an even functional, the subspaces and are defined by (3.13).Suppose that, for every , there exist such that(A2) as ,(A3),(A4) satisfies condition for every .
Then, has a sequence of critical values tending to .
Proposition 3.5 (see ). Suppose that hypotheses , and if , denote
Let us introduce the following lemma that will be useful in the proof of our main result.
Lemma 3.6. Let , and assume that are satisfied, then satisfies (PS) condition.
Proof. By Propositions 2.2 and 2.3, we know that if we denote then is weakly continuous and its derivative operator, denoted by , is compact. By Propositions 3.1 and 3.2, we deduce that is also of (S+) type. To verify that satisfies (PS) condition on , it is enough to verify that any (PS) sequence is bounded. Suppose that such that Then, for large enough, we can find such that Since , we have . In particular, is bounded. Thus, there exists such that We claim that the sequence is bounded. If it is not true, by passing a subsequence if necessary, we may assume that . Without loss of generality, we assume that appropriately large such that for any . From (3.18) and (3.19) and letting , then , we have By virtue of assumptions and and combining (3.20) and (3.21), we have Note that , let we obtian a contradiction. It follows that the sequence is bounded in . Therefore, satisfies (PS) condition.
Under appropriate assumptions on the perturbation terms , a sequence of weak solutions with energy values tending to was obtained. The main result of the paper reads as follows.
Theorem 3.7. Let , and , and assumed that are satisfied; then has a sequence of critical points such that as .
Proof. We will prove that satisfies the conditions of Proposition 3.4. Obviously, because of the assumptions of and , is an even functional and satisfies (PS) condition (see Lemma 3.6). We will prove that if is large enough, then there exist such that (A2) and (A3) hold. By virtue of , , there exist two positive constants such that
Letting with appropriately large such that , we have
If , then by Proposition 3.5, we have
Choose . For with , we have
Since as and , we have and . Thus, for sufficiently large , we have with and as . In other cases, similarly, we can deduce
So (A2) holds.
By virtue of and , there exist two positive constants such that Letting , we have If , then we have Since , all norms are equivalent in . So we get Also, note that , Then, we get as . 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 . Thus, we complete the proof.
This time our idea is to show that possesses a nontrivial global minimum point in .
Theorem 3.8. Let , and assume , are satisfied; then (1.1) has a weak solution.
Proof. Firstly, we show that is coercive. For sufficiently large norm of , and by virtue of (3.23), If then So is coercive since . Secondly, by Proposition 2.2, it is easy to verify that is weakly lower semicontinuous. Thus, is bounded below and attains its infimum in , that is, and is a critical point of , which is a weak solution of (1.1).
In the Theorem 3.8, we cannot guarantee that is nontrivial. In fact, under the assumptions on the above theorem, we can also get a nontrivial weak solution of .
Corollary 3.9. Under the assumptions in Theorem 3.8, if one of the following conditions holds, (1.1) has a nontrivial weak solution.(1)If , there exist two positive constants such that (2)If , there exist two positive constants such that (3)If , there exist two positive constants such that
Proof. From Theorem 3.8, we know that has a global minimum point . We just need to show that is nontrivial. We only consider the case here. From (1), we know that for small enough, there exists two positive constants such that Choose ; then . For small enough, we have Since and , there exists small enough such that . So the global minimum point of is nontrivial.
Remark 3.10. Suppose that and ; then the conditions in Corollary 3.9 can be fulfilled.
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