On Elliptic Equations With Discontinuous Nonlinearity And Obstacle Problem

In this paper, we mainly study the existence of nontrivial solutions of elliptic equation with discontinuous nonlinearity and the obstacle problem with generalized minimax principle.This paper is divided into three parts. The first part is mainly devoted to generalize some minimax theorems. Basing on the Deformation Lemma, we generalize some minimax theorems for a class of special functional, which is Lipschitz continuous. At the same time, we give out an application of the generalized mountain pass theorem. We discuss the existence of the nontrivial solution of the equation ? ?u ?λu=f( x,u) with Dirichlet zero boundary value, where ?∈Rn is a bounded field with smooth boundary. f ( x,t) is a locally bounded measurable function defined on ?×R. We prove, under proper conditions onλand f ( x,t), the above Dirichlet problem has at least a nontrivial generalized solution. In the process of the discussion, the generalized gradient is a basic tool.In the second part, we give out an application of the generalized mountain pass lemma to solve another elliptic equation with discontinuity nonlinearity. Firstly, we introduce some notions and results related to the Orlicz-Sobolev space, then discuss the existence of the nontrivial solution of the equation ? div ( a(|?u|)?u)=g(x,u) with Dirichlet zero boundary value in an Orlicz-Sobolev space. We prove, under proper condition on a (s) and the nonlinearity g ( x,t), the above Dirichlet problem has at least a nontrivial generalized solution.In the third part, we study the existence of the nontrivial solution of the following obstacle problem where K = {v∈H 01( ? ): v≤ψa. e . ?} , ? is an open bounded field in R n, and its boundary is smooth. We prove, under proper conditions on the obstacleψ(x), a (x) and the nonlinearity p ( x,t), the above Dirichlet problem has a nontrivial solution.