The Existence Of Nontrivial Solutions For Kirchhoff Type Problems In R~N

In this paper,firstly,the existence of positive bound solutions for Kirchhoff type problem in R3(?)is investigated by using variational methods,a new version of global compactness lemma and the linking theorem.where a and b are two positive constants,p ?(1,5)and V:R3 ? R is a potential function satisfying:(V1)there exists a positive constant V? such that for all x ? R3,(?)The main results is as follows.Theoreml.If one of the following conditions is satisfied,then the problem(P1)possesses no ground state solution.(1)p ?(1,3]and V satisfies(V1),(V2);(2)p ?(3,5)and V satisfies(V1).Theorem2.If p ?(3.5)and V satisfies(V1),Moreover,when(?)is small enough,then problem(P1)admits at least a positive bound state solution.Secondly,the existence of nontrivial solutions for p-Kirchhoff type equations in RN with critical nonlinearities is investigated by using variational methods combined with the mountain pass lemma and vanishing lemma.where a,b,m>0 are positive constants,q?(p,p*),p?(1,N),p*=Np/N-p,N ? 3,?>0 is parameter,p-Laplacian operator?pu:=div(|?u|p-2?u).The main result is as follows.Theorem3.Assume(?)then when one of the following conditions hold,the equation(P2)has at least one nontrivial solution.(?)is sufficiently large.The structure of this paper is as follows.In the first chapter,the research background of Kirchhoff type problems is briefly described,and the research work and research results in this paper are stated.In the second chapter,the necessary knowledge for proving the existence of the positive bound state solution of(P1)and the nonexistence of the basic state solution is given,and the proof process of the main results(theorem 1,theorem 2)is given by them.In the third chapter,the preparatory knowledge needed to prove the existence of nontrivial solutions of the problem(P2)is given,and the proof process of the main result(theorem 3)is given by using them.