Existence And Multiplicity Of Solutions For A Class Of Elliptic Equations Involving Critical Sobolev-Hardy Exponents And Hardy Potential

This thesis investigates a class of elliptic equations with critical Sobolev-Hardy exponents and Hardy potentialWhere N≥3,0≤s<2,0≤μ<μ:=(N-2/2)2,2*(s):=2(N-s)/N-2is the critical sobolev-Hardy exponent.The functions h(x),Q(x),k(x)and the parameter q satisfy some assumptions respectively.We consider the solutions of the equations with the following two cases.The first case:when Q(x)三1,h(x)≠0,that is’By using variational methods including Mountain Pass Theorem,concentration-compactness principle,we prove the existence of nontrivial solutions for the equation.And by the maximum principle,we obtain the positive solutions of the equation. The second case:considering when h(x)=0, Q(x)≠0, that isBe similar to the case one, by employing the variational methods and analytic tech-niques, we obtain the existence and multiplicity of the solutions for the equation. This thesis consist of three chapters. The first chapter is devoted to discuss the introduction including research background, research situation, research achievements and symbols instruction. The second chapter deals with the existence of the positive solutions for the equation under the first case, the main conclusion is Theorem2.1.1. In the last chapter we research the the existence and multiplicity of the solutions for the equation about the second case, the main results are Theorem3.1.3and3.2.5.