Multiple Solutions For A Class Of Semilinear Elliptic

In this thesis, we will be concerned with the multiple solutions for the following semilinear elliptic variational inclusion problem with a nonsmooth potential: whereΩ(?) RN (N≥3) is a bounded domain with a smooth boundary (?)Ω, j(x,ζ):Ω×R→R, j(x,·) is a locally Lipschitz function, (?)j(x,ζ) is the generalized gradient, c(x)∈L∞(Ω). The problem (P) is also called the hemivariational inequality problem.Recently many authors have studied elliptic variational problems with a smooth or nonsmooth potential by variational methods and critical point theory. In problem (P), when c(x) is an eigenvalue of (-△,H01(Ω)), problem (P) is called the resonance problem. The existence and multiple solutions of such problems have been studied by Z. Denkowski, L. Gasinski and N. S. Papageorgiou in [1, 2, 3] by using the local linking theory. In this paper, we mainly consider the multiple solutions of the non-resonance problem (when c(x) lies between the two successive eigenvalues). N. S. Papageorgiou studied the existence of a nontrivial solution of this non-resonance problem by general minimax principle in [4].In this paper, we first consider the semilinear elliptic variational inclusion problem (P) on a finite dimensional space by using the reduction method, then we obtain that there are at least two nontrivial solutions for problem (P) at non-resonance by using the local linking theory.