Existence For Some Elliptic Equations And Systems

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, we shall investigate the existence of weak solutions for following problem: Using Hardy's Inequalities can define the equivalent norm. Verify the geometic condi-tions of the Mountain Pass Lemma we get the existence of a nontrivial solution for a class of p-Laplacian equation with indefinite weights.Chapter three studies the existence of nontrivial solutions for a class of variational inclusions system of form: where N≥2,2≤p≤N, F:R2→R is a locally Lipschitz not necessary smooth potential function and△pu=div(|▽u|p-2▽u). We denote by (?)F(u,v) the partial generalized gradient of F(·,v) at the point u and by (?)F(u,v) that of F(u,·)at v. There are two difficulties in treating the system. Firstly, the energy functional F we consider is no smooth and due to unbounded domain the compactness is lost. Secondly, the topological dual E* of E is no equivalent to E and how to control the growth of the nonlinearity F. The tools of the nonsmooth critical point theory come from the subdifferential theory for locally Lipschitz functions due to Clarke and the Principle of Symmetric Criticality of Krawcewicz and Marzantowicz which will be crucial in this dissertation.