The Existence And Multiplicity Of Nontrivial Solutions For Two Kinds Of -Kirchhoff Type Equations In R~n

In this paper,we investigate the existence of solutions for two kinds of p-Kirchhoff type equations in RN by using variational methods.Firstly,we study the existence and multiplicity of nontrivial solutions for p-Schrodinger-Kirchhoff type equations with subcritical nonlinear term.Secondly,we study the existence of positive solutions for p-Kirchhoff type equations with critical nonlinear term.For the p-Schrodinger-Kirchhoff type equations where a,6>O are constants,1<p<N,and the nonlinear term f is subcritical.The assumptions are as follows:then we have the following results.Theorem 1 Under the assumptions(V),(f1)and(f2),the problem(Pi)possesses at least a nontrivial solution.Theorem 2 Under the assumptions(V)and(f1)-(f3),the problem(P1)possesses infinitely many nontrivial solutions.For the p-Kirchhoff equations where a,b>0 are constants,1<p<N,and the nonlinear term f is critical.The assumptions are as follows:then we have the following result.Theorem 3 For p2<p*,and under the assumptions(f1)-(f5),if one of the following conditions is satisfied,then the problem(P2)possesses at least a positive solution.The structure of this paper is as follows.In the first chapter,we introduce the background of Kirchhoff type problems and describe the research contents and results obtained in this paper.In the second chapter,we firstly give some preliminaries which will be used to prove the existence and multiplicity of nontrivial solutions for the problem(P1).Then we give the proof of the main results.In the third chapter,we firstly give some preliminaries which will be used to prove the existence of solutions for the problem(P2).Then we give the proof of the main result.