The Qualitative Research Of Nonlinear Kirchhoff Type Elliptic Boundary Value Problems

In this thesis,the Kirchhoff elliptic boundary value problem with Sobolev-Hardy critical exponent is studied.The existence and multiplicity of positive solutions of Kirchhoff elliptic equations with Sobolev-Hardy critical exponent are discussed,as well as the existence of positive solutions of Kirchhoff elliptic systems with Sobolev-Hardy critical exponent are investigated.First,a class of Kirchhoff elliptic equations with Sobolev-Hardy critical exponent is studied.There are two difficulties in studying this equation:On the one hand,the equation itself is no longer an equation in point-by-point sense because it contains nonlocal terms;On the other hand,the equation contains the Sobolev-Hardy critical exponent,which makes the Sobolev embedding(?)lose its compactness.Therefore,the energy functional corresponding to this problem no longer satisfies the Palais-Smale condition,and the usual variational method and related analytical techniques cannot be used.This thesis overcomes the problem of lack of compactness by applying Lions concentrated compactness principle.It is proved that the energy functional of the equation satisfies the local Palais-Smale condition by some analytical techniques.Finally,the existence and multiplicity of the solution are proved by the mountain lemma and the strong maximum principle.Secondly,a class of Kirchhoff elliptic coupling systems with strong coupling terms and Sobolev-Hardy critical exponent are investigated.The main difficulty in solving the problem is that the system contains both nonlocal term and Sobolev-Hardy critical exponent,and strong coupling term,which makes the problem more complex than the general equation.This thesis uses Brezis-Lieb lemma to overcome the lack of compactness.Nehari manifold,fiber mapping and other methods are used to prove that the energy functional of the system has a critical sequence on the level set c,and it is proved that the energy functional satisfies the Palais Smale condition.Finally,the existence of the positive solution of the system is obtained by using the strong maximum principle.Finally,the main research content of this thesis is summarized,and some existing research problems and the specific direction of future research are put forward.