Positive Solutions To A Kirchhoff Type Problem With Parameter

In the present paper, we firstly study the existence of positive solutions to Kirch-hoff type problem with a parameter by using variational method and iterative technique, Secondly, the existence of positive solutions to Kirchhoff type problem with Hardy term is investigated by using variational method and priori estimate.Firstly, we study the existence of positive solutions to Kirchhoff type problem with a parameter where N≥3, a is a positive constant, λ≥ 0 is a parameter, m,f:R+â†'> R+ are continuous functions, and Q is a potential function.To the Kirchhoff type problem with a parameter, Firstly, through fixing the non-local term the Kirchhoff type equation is changed into elliptic type equation, the cor-responding functional Jω with the ellipse equation is obtained by using variational method, and the functional is proved to satisfy the geometric construction of the Lem-ma proposed by the paper, and then there exists a (PS) sequences of functional Jω. Without usual compactness conditions, a Pohozaev type identity is utilized to obtain the boundedness of the (PS) sequences. Then, it proves that the bounded (PS) sequences has a strong convergent subsequences and the subsequences convergs to a nontrivial critical point uω of functional Jω. Finally, according to iterative method it obtains the critical point sequences{un} of functional{JUn-1} via the iteration of ω, and{un} is confirmed to converge to a positive solution of the Kirchhoff type equation. It draws the conclusion that there exists at least one positive solution when the parameter is in a certain range.Secondly, we investigate the existence of positive solutions to a Kirchhoff type problem with Hardy term where N≥3, a is a positive constant, μ, λ≥0 are parameters,f :R+â†'R+ is a continuous function.To the Kirchhoff type problem with Hardy term, Firstly, the corresponding func-tional with the Kirchhoff problem is obtained by using variational method, combining Hardy-embedding theorem and Sobolev-embedding theorem, the functional is proved to satisfy the geometric construction of the Lemma proposed by the paper, and then there exists a (PS) sequences of the functional. Without usual compactness conditions, a Pohozaev type identity is used to obtain the boundedness of the (PS) sequences, a priori estimate proves that the bounded (PS) sequences has a strong convergent subse-quences and the subsequences convergs to a nontrivial critical point of the functional. It draws the conclusion that there exists at least one positive solution when the parameter is in a certain range.This paper consists of four chapters. The structure is as follows:In the first chapter, we introduce the background of the discussed problem and give the main results of this thesis.In the second chapter, we give some necessary preliminaries of the variational method of this paper.In the third chapter, firstly, we give the variational structure of Kirchhoff problem with a parameter, that is the corresponding energy functional of the problem. Then, by using variational method, iterative technique and embedding theorem, we prove the existence of positive solutions to the Kirchhoff type problem.In the fourth chapter, firstly, we give the variational structure of Kirchhoff problem with Hardy term, that is the corresponding energy functional of the problem. Then, by using variational method, combining priori estimate and embedding theorem, we prove the existence of positive solutions to the Kirchhoff type problem.