Existence Of Positive Solutions For Two Kinds Of Kirchhoff Type Equations

The existence of positive solutions for Kirchhoff type equation with multiple critical nonlinearities in R3 and the existence of positive ground state solutions for p-Kirchhoff type equation in-RN are investigated by using variational methods. The main theoretical bases are Mountain pass lemma and the method of combining Pohozaev manifold with Nehari manifold.Firstly, we consider the following Kirchhoff type equation with multiple critical nonlin-earities: where a, b> 0 are constants,2*=6,2* (s)=6-2s, s∈(0,1).The main result is as follows.Theoreml. For any a, b> 0, s∈(0,1), problem (P1) has at least a positive solution in D1,2 (R3).Secondly, we investigate the following p-Kirchhoff type equation: where a> 0, b≥0 are constants, V∞> 0,1<p, q< N, p*=Np/N-p.△p is the p-Laplace operator.We assume that p, q satisfy(p1) p∈(1,N), q ∈(p,p*), q∈(p2(1+N-p)-Np,p*),The main results are as follows.Theorem2. When N=3, p,q satisfy (p1), problem (P2) has at least a positive ground state solution.Theorem3. WhenN≥ 4,p ,q satisfy(2) or(3), problem(2) has at least a positive ground state solution.The structure of this paper is as follows.In the first chapter, the essential ideology of variational methods and the recent development of Kirchhoff type problems by variational methods are introduced. Moreover, the research work and main results of this paper are stated.In the second chapter, the preliminaries to prove the existence of positive solutions for Kirchhoff type equation with multiple critical nonlinearities in R3 are presented. More importantly, the proof of the main result is given.In the third chapter, the preliminaries to prove the existence of positive ground state solutions for -Kirchhoff type problem in Rare presented. More importantly, the proofs of the main results are given.