Existence Of Ground State Solutions And Nonnegative Solutions For Two Kinds Of Elliptic Problems

Variational method,one of the basic methods to solve the boundary value problem of differential equations,transforms the solution of differential equations into the critical point of corresponding energy functional in the function space.In this paper,due to the lack of compactness of functional,the solution of the equation cannot be obtained directly by using classical variational method.For this reason,the existence of ground state solutions and nonnegative solutions for two kinds of elliptic problems with critical terms is proved by using variational method,Nehari manifold and mountain path theorem.The thesis consists of three sections.Chapter 1 is the preface.In Chapter 2,we consider the following semilinear elliptic problem with Hardy-Sobolev critical termwhere N?3,0??,??? is a smooth bounded domain with boundary,????[0,???),???=?N-2?2/4,s?[0,2),2*?s?=2?N-s?/?N-2?is the Hardy-Sobolev critical exponent,the nonlinear term f satisfies?f0?f?L^r?Q?,m?{x??:f?x?>0}?>0,where r=2*/?2*-q?.NoteThe conclusions are as follows:Theorem 2.1.1.If q??1,2?and?f0?holds,problem?0.3?exists a ground state solution ???or any ???0,?₁?.Theorem 2.1.2.If q=1 and?f0?holds,problem?0.3?exists a nonnegative nontrivial solution ???or any ???0,?₂?.In Chapter 3,we consider the ???ollowing nonlocal elliptic problem with Hardy-Sobolev critical termwhere N??4?,a,b>0,0??,????R^N is a smooth bounded domain with boundary,????[0,???),???=?N-2?2/4,s?[0,2),2*?s?=2?N-s?/?N-2?is the Hardy-Sobolev critical exponent,the nonlinear term F satisFies????0?F?L^r???,there exists u₀?H??????such that(q?u₀?^q)^-1 ???f??u₀⁺?^q>?2a+b?/4,where r=2^*/?2^*-q?.NoteThe conclusions are as follows:Theorem 3.1.1.If q=2 and????0?holds,problem?0.4?exists at least a nonnegative nontrivial solution ???or any ???0,?₃?.Theorem 3.1.2.If q??2,2*?and????0?holds,problem?0.4?exists at least two non-negative nontrivial solutions for any ???0,?₃?.