The Existence And Multiplicity Of Nontrivial Solutions For P-kirchhoff Type Equations In Total Space

In this paper,the existence and multiplicity of nontrivial solutions for p-Kirchhoff equations in RN are investigated by using variational methods.Firstly,the existence of infinitely many small solutions for the p-Kirchhoff type problem(?) which has a local sublinear nonlinearity is studied by using the generalized form of Clark's theorem combined with the Moser iteration,where a and b are two positive constants,1<p<N,N ? 3,?pu =div(|?u|p-2?u),is the p-Laplace operator,the hypotheses of V and f are as follows:(V)V?C(RN,R),and(V+)-1/p-1 is integrablo near infinity,i.e.,(f1)There exists ?>0 such that f ?C(RN ×[-?,?],R),and f(x,-t)=-f(x,t)for all |t| ? ? and x ? RN;(f2)lim u?0(?0uf(x,s)ds/|u|p=+? uniformly in some ball Br(x0)(?)RN;(f3)There exist ?>0 and C>0 such that |f(x,t)|? C for all |t|<p and x? RN.The main result is as follows.Theorem 1 Under the assumptions(V)and(f1)-(f3),(P1)possesses a sequence of solutions {un} such that |un|??0 as n??.Secondly,the existence of nontrival solutions for the p-Kirchhoff type problem(?) which has a critical nonlinearity is studied by using variational methods combined with the mountain pass lernmma and the cormpactness principle,where a,b>)are constants,1<p<N,N?3,p<q<p*,p*=Np/N-p,parameter ?>0,k satisfies the following assumptions:(k1)k?Lp*/p*-q(RN),k(x)?0 and k(x)(?)0,(?)x?RN;(k2)There exist x0?Rn,?,?1>0,0<?<N,such that k(x)??|x-x0|-? as x?B?1(x0)\{x0}.The main result is as follows.Theorem 2 Assume(?) and(k2)are satisfied,then there exist constants ?*,?**>0 such that when one of the following conditions is satisfied,(P2)possesses at least one nontrivial solution.The structure of this paper is as follows.In the first chapter,we introduce the research background of Kirchhoff equation and the new achievements in recent years,and state the research contents and results obtained in this paper.In the second chapter,the necessary preparatory knowledge to prove the existence of infinitely many small solutions for the problem(P1)is presented,and the proof process of the main result is given.In the third chapter.the necessary preparatory knowledge to prove the existence of nontrivial solutions for the problem(P2)is presentedt and the proof process of the main result is given.