The Study Of Hemi-Variational Inequality

In this paper we deal with solutions of existence theorems for several case of hemivariational inequality.In the first chapter, we introduce some definitions, lemmas and background of hemivariational inequality.In the second chapter, we study the problem of the following p-laplacian equation with Dirichlet boundary .WhereΩ∈R~n is a bounded domain , the potential function j(z, ?) is locally Lipschitz. The problem is studied by so many scholars with different methods. Such as topology degree theory, mountain pass lemma, linking theorem and so on. Using the above the-orem, the first thing we need to do is proving nonsmooth PS compactness condition. In the compactness condition, we already had the generalized nonsmooth PS compactness condtion. This chapter we will prove the existence of a theorem through the generalized nonsmooth PS compactness condition.In the third chapter, we study the problem of the following p(x)-laplacian equation with Dirichlet boundary .The p(x)-laplacian possesses more complicated nonlinearity than the p-laplacian, for example, it is inhomogeneous and in general the infimum of the eigenvalues of p(x)-laplacian equals to 0, so it is more difficult to deal with. Through our assumptions for the nonsmooth potential and also for p(x), using linking theorem of the nonsmooth critical theorem, we prove a multiplicity result.In the fourth chapter, we concerned with the problem of quasilinear hemivaria-tional inequality, to study radial and unradial solutions.The radial and unradial solutions of seniilinear hemivariational inequality had been obtained. so this chapter, we consider quasilinear. Using the fountain theorem and mountain theorem with nonsmooth condition, we prove the existence of an nontrivial solution, as well as infinitely many radially and unradially solutions. In the proofs we use the principle of symmetric criticality for locally Lipschitz functions.