Existence Of Nontrivial Solution For Elliptic Equations With Hardy Potential

In this paper,we mainly study existence of nontrivial solution for elliptic partial differential equations with hardy potential and critical growth condition,and a nontrivial solution is given in W01,p(Ω). Handling of Hardy potential and weak continuity of Fqi(x,u,Du), Fu(x,u,Du) are the primary process of theorem proving. First,we proof that the associated energy functional I(u) satisfy some geometry condition by means of critical growth condition of F(x,u,Du) and hardy inequality,and get some results for I(u) by Mountain Pass Lemma without (PS) condition,that is, existing bounded sequence{un}of W01,p(Ω),such that:I(un)→c0;I'(un)→0 in(W01,p(Ω))*. Then,we explain that Sobolev imbedding W01,p(Ω)→Lq(Ω) lose compactness at most countable point for subsequence by concentration compact-ness theory.Next taking out this most countable point singularity, proving Fqi(x,u,Du), Fu(x,u,Du)are weak continuity for subsequence,and give result of existence of nontriv-ial solution for problem.Finally,we study existence of nontrivial solution under changing critical growth condition,and proof the associated energy functional I(u) satisfy some geometry condition, weak continuity of Fqi(x,u,Du), Fu(x,u,Du),and existence of non-trivial solution.