Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by the -Laplacian

The Scientific World Journal, Dec 2013

A class of nonlinear Neumann problems driven by -Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions.

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Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by the -Laplacian

Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by the -Laplacian Qing-Mei Zhou Library, Northeast Forestry University, Harbin 150040, China Received 12 September 2013; Accepted 5 December 2013 Academic Editors: A. Atangana, A. Kılıçman, S. C. O. Noutchie, and S. S. Ray Copyright © 2013 Qing-Mei Zhou. 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. Abstract A class of nonlinear Neumann problems driven by -Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions. 1. Introduction Recently, there are several papers on the research of the Neumann-type problems involving the -Laplacian. Of the existing works in the literature, the majority deal with problems in which the potential function is smooth (i.e., ). We mention the works of Mihailescu [1], Fan and Ji [2], Yao [3], Shi and Ding [4] and Cammaroto et al. [5]. Problems with a nonsmooth potential, were studied by Dai [6, 7], who for the case established the existence of three or infinitely many solutions for Neumann-type differential inclusion problems involving the -Laplacian, using the nonsmooth three-critical-points theorem and nonsmooth Ricceri type variational principle, respectively. Not long ago, Qian et al. [8] studied the nonhomogeneous Neumann problem with indefinite weight; that is, where is a bounded domain with smooth boundary , , is a function possibly changing sign, is the trace operator with for all , and are locally Lipschitz functions in the -variable integrand (in general it can be nonsmooth), and are the subdifferentials with respect to the -variable in the sense of Clarke [9]. The authors prove the existence of at least one nontrivial solution of (1) using the nonsmooth Mountain Pass theorem and Weierstrass theorem. If , then problem (1) becomes problem (2) as follows: In this paper, our goal is to establish the existence of at least two nontrivial solutions for problem (2). We emphasize that the operator is said to be -Laplacian, which becomes -Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearities than the -Laplacian; for example, it is inhomogeneous and in general, it has not the first eigenvalue. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications to modeling electrorheological fluids (see [10, 11]) and image restoration (see [12]). This paper is divided into three sections: in the second section, we introduce some necessary knowledge on the nonsmooth analysis and basic properties of the generalized Lebesgue-space and the generalized Lebesgue-Sobolev space . In the third section, we give the assumptions on the nonsmooth potentials ,   and prove the multiplicity results for problem (2). 2. Preliminary In this section, we first review some facts on variable exponent spaces and . For the details, see [13–18]. Firstly, we need to give some notations, which we will use through this paper: Obviously, . Denote by the set of all measurable real functions defined on . Two functions in are considered to be one element of , when they are equal almost everywhere. For , define with the norm with the norm Denote Let be a Banach space and its topological dual space and we denote as the duality bracket for pair . A function is said to be locally Lipschitz, if for every , we can find a neighbourhood of and a constant (depending on ), such that , for all . For a locally Lipschitz function , we define It is obvious that the function is sublinear and continuous and so is the support function of a nonempty, convex, and -compact set , defined by The multifunction is called the generalized subdifferential of . If is also convex, then coincides with subdifferential in the sense of convex analysis, defined by If , then . A point is a critical point of if . It is easily seen that if is a local minimum of , then . A locally Lipschitz function satisfies the nonsmooth -condition at level (the nonsmooth -condition for short), if for every sequence , such that and , as , there is a strongly convergent subsequence, where . If this condition is satisfied at every level , then we say that satisfies the nonsmooth -condition. Lemma 1 (see [19]). Consider the following.(1)The spaces and are separable and reflexive Banach spaces. Moreover, is uniform convex.(2)If and for any , then the imbedding from to is compact and continuous.(3)If and for any , then the imbedding from to is com (...truncated)


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Qing-Mei Zhou. Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by the -Laplacian, The Scientific World Journal, 2013, 2013, DOI: 10.1155/2013/753262