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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 236712, 10 pages
Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz-Sobolev Spaces
Department of Mathematics and Computational Sciences, Hengyang Normal University, Hengyang, 421008 Hunan, China
Received 14 July 2012; Accepted 31 August 2012
Academic Editor: Juntao Sun
Copyright © 2012 Liu Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz-Sobolev space.
Consider the following nonhomogeneous Neumann problem with a perturbed term: where is a bounded domain in with smooth boundary , is the outer normal to , are two Carathéodory functions, are two parameters, and the function is such that defined by is an odd, strictly increasing homeomorphism from to .
It is well known that these kinds of problems are important in applications in many fields, such as elasticity, fluid dynamics, and image processing (see [1–4]). Since the operator in the divergence form is nonhomogeneous, we introduce Orlicz-Sobolev space which is an appropriate setting for these problems. Such space originated with Nakano  and was developed by Musielak and Orlicz . Many properties of Sobolev spaces have been extended to Orlicz-Sobolev space (see [7–10]). Several authors have widely studied the existence of solutions for the relevant problem by means of variational techniques, monotone operator methods, fixed point, and degree theory (see [11–15]). To the best of our knowledge, for the perturbed nonhomogeneous Neumann problem, there has so far been few papers concerning its multiple solutions. Motivated by the above facts, in this paper, we establish some new sufficient conditions under which such a problem possesses three weak solutions in Orlicz-Sobolev space.
We start by recalling some basic facts about Orlicz-Sobolev space. Let be as in Introduction and , We observe that is, a Young function, that is , is convex and . Furthermore, since if and only if , , and , then is called an -function. The function is called the complementary function of and it satisfies Assume that satisfies the following structural hypotheses ; . Further, we also assume that the function is convex.
The Orlicz space defined by is the space of measurable functions such that Then is a Banach space whose norm is equivalent to the Luxemburg norm We denote by the Orlicz-Sobolev space, defined by This is a Banach space with respect to the norm
Lemma 2.1 (see ). On the norms are equivalent. Moreover, for every , one has
Lemma 2.2. Let , then
Proof. For the proof of
we can see Lemma 2.2 of the paper . Since for all , it follows that letting , we have
Thus, one has
Moreover, by the definition of the norm, we remark that
Therefore, we have
for all .
Similar techniques as those used in the proof of (2.12), we have Therefore, we can obtain for all .
Lemma 2.3 (see ). Let and assume that for some , then one has .
Lemma 2.4 (see ). If , then is compactly embedded in and there exists a constant such that where .
Now, one recall, a three critical theorem of B. Ricceri. If is a real Banach space, denote by (see ) the class of all functionals possessing the following property: if is a sequence in converging weakly to and , then has a subsequence converging strongly to . For example, if is uniformly convex and is a continuous, strictly increasing function, then, by a classical results, the functional belongs to the class .
Lemma 2.5 (see ). Let be a separable and reflexive real Banach space; let be a coercive, sequentially weakly lower semicontinuous functional, belonging to , bounded on each bounded subset of and whose derivative admits a continuous inverse on ; a functional with compact derivative. Assume that has a strict local minimum with . Finally, setting assume that . Then for each compact interval (with the conventions , there exists with the following property: for every and every functional with compact derivative, there exists such that, for each , the equation has at least three solutions in whose norms are less than .
Lemma 2.6 (see ). Let be a reflexive real Banach space; an interval, let be a sequentially weakly lower semicontinuous functional, bounded on each bounded subset of and whose derivative admits a continuous inverse on ; a functional with compact derivative. Assume that for all , and that there exists such that Then there exist a nonempty open set and a positive number , with the following property: for every and every functional with compact derivative, there exists such that, for each , the equation has at least three solutions in whose norms are less than .
Lemma 2.7 (see ). Let be a nonempty set and two real functions on . Assume that there are and such that Then for each satisfying one has
3. Proof of the Main Results
Theorem 3.1. Let be a function satisfying the structural hypotheses – and the following conditions hold, . Then, for each compact interval , there exists with the following property: for every and , there exists such that, for each , the problem has at least three weak solutions whose norms in are less than .
Proof of Theorem 3.1. In order to apply Lemma 2.5, we let
We divide our proof into two steps as follows.
Step 1. We show that some fundamental assumptions are satisfied.
. Obviously, is a separable and reflexive real Banach space (see ). By Lemma 2.2, it is easy to see that is a coercive, bounded on each bounded subset of . On the other hand, with the derivatives given by for any . Hence, the critical points of the functional are exactly the weak solutions for problem . Moreover, owing that is convex, it follows that is convex. Hence, one has that is sequentially weakly lower semicontinuous. The fact is compactly embedded into implies that operators is compact. As the proof of Lemma 3.2 in , we know that has a continuous inverse.
Moreover, if is a sequence in converging weakly to and , remark that is sequentially weakly lower semicontinuous, one has Then, up to a subsequence, we deduce that . Taking into account that converges weakly to and is sequentially weakly lower semicontinuous, we have We assume by contradiction that does not converge to in . Hence, there exist and a subsequence of such that Then there exists such that On the other hand, (see ). Letting in the above inequality we obtain and that is a contradiction with (3.4). It follows that converges strongly to and . In addition, .
Step 2. We show that .
In view of , for all , there exists such that for any . For , we have By , for all , there exists such that for any . Further, for each , we have So we get Then, with the notation of Lemma 2.5, we have . By assumption , we have . Thus, all the hypotheses of Lemma 2.5 are satisfied. Clearly, . Finally, by Lemma 2.5, we can obtain the Theorem 3.1.
Remark 3.3. Since in  is -sublinear, the results of  do not fit to the problem treated in the previous Example 3.2 even if , that is, there is no perturbed nonlinear term. In addition, for nonhomogeneous Neumann problem with a perturbed term, we can have the following result when is -sublinear, which extends the results of .
Theorem 3.4. Let be a function satisfying the structural hypotheses – and the following conditions.
There exist two constants with such that and where denotes the Lebesgue measure of the set .
There exist and such that for every .
Then, there exist a nonempty open set and a positive number , for each and for every , there exists such that, for each , the problem has at least three weak solutions whose norms in are less than .
Proof of Theorem 3.4. Let us consider as the proof of Theorem 3.1. For any , by we have
for . Since , one has for all .
Let , . For , we have Hence, one has From , it follows that Since all the assumptions of Lemmas 2.7 and 2.6 are satisfied, then, there is a nonempty open set and a positive number , for each and for every there exists such that, for each , the problem has at least three weak solutions whose norms in are less than .
L. Yang was supported by the Natural Science Foundation of Hunan Province (12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117), and Hengyang Normal University Doctoral Science Foundation (11B35).
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