Existence And Multiplicity Results For Some Elliptic Hemivariational Inequalities

This dissertation collects the main results obtained by the author during the period when she has applied for the M.D.The contents are the following:In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.Chapter three studies the multiplicity of positive solutions for the following Dirich- let problem involving p(x)-Laplacian operator.The main difficulty is that the p(x)-Laplacian operator possesses more complicated nonlinearities than the p-Laplacian. To overcome this difficulty, our approach is variational based on critical point theory, and the multiplicity of positive solutions is proved by using some special techniques.Chapter four deals with the existence result for the problem about the Schrodinger equation with nonsmooth potential in unbounded domain, that is:where V > 0, it is a continuous periodic potential function, the function j(x,u) is locally Lipschitz in u. Since the solutions of this problem are the critical points of the associated energy function. One generally needs some compactness such as PScondition or C-condition to prove the existence of critical points of the energy func-tion, but when we study the elliptic equation in R^N, the compactness condition does not always hold since the imbedding of the Sobolev space H₀^1,2(Ω)into L^2*(Ω) is not compact. In this chapter, based on the nonsmooth critical point theory, and by using the approximation technique with periodic function, the existence of nontrivial solution is obtained.