On Qualitative Research For The Critical Nonlinear Elliptic Boundary Value Problems Involving Negative Exponent Terms

This thesis studies the critical nonlinear elliptic boundary value problems with negative terms.This thesis discusses the existence and multiplicity of positive solutions for a class of quasilinear elliptic equations involving Sobolev critical exponent term and negative exponent term.Then,the existence of positive solution for a class of semilinear elliptic systems involving Sobolev critical exponent and negative exponent is studied.First,a class of quasilinear elliptic equations with Sobolev critical exponential term and negative exponent term is studied.Due to the existence of Sobolev critical exponents,the Sobolev embedding (?) W ì W is not compact.The negative exponential causes the corresponding energy functional of the problem not Frechet differentiable,which makes it impossible to apply critical point theory to deal with such problems.To solve these difficulties,a Nehari set which contains all weak solutions of the problem is established,and the corresponding functional is proved to be lower bounded in the Nehari set.Secondly,Brezis-lieb lemma and Vitali theorem are used to prove the existence of a solution whose corresponding functional value is a local minimum,and the strong maximum principle is used to prove that it is a positive solution.Finally,the concentration compactness principle due to Lions and Ekeland’s variational principle are used to prove the existence of another weak solution of the equation,and the strong maximum principle is used to prove that the weak solution is a positive solution.In this chapter,the existence and multiplicity of the positive solutions of the equation are proved.Then,a class of semilinear elliptic coupled systems with Sobolev critical exponential terms and negative exponential terms is studied.The main difficulty to solve the problem lies in the lack of compactness of Sobolev embedding (?).The existence of negative exponential term makes the functional not Frechet differentiable,so()cPS sequence corresponding to the functional can not be found.By using Nehari set,Ekeland variational principle and Vitali theorem,these difficulties are overcomed.The positive ground state weak solution of the problem is proved to be existed.At last,the main contents of this thesis and put forward the existing problems and future research directions is summarized.