Least Energy Solution For A Nonlinear Kirchhoff Type Elliptic Equation

In this paper,we study the following nonlinear Kirchhoff type elliptic equation-(a+b?R3|?u|2dx)?u+V(x)u=?u+|u|p-1u,x?R3,u?H1R3),(0-1)where a,b>0 are constants,?>0,p?(3,5),V(x)?C(R3,R+)and(V1)holds:V(x)??as |x|??.It is clear that weak solutions to(0-1)correspond to critical points of the energy functional I?(u)=1/2?R3a|?u|2+V(x)u2dx+b/4(?R3|?u|2dx)2-?/2?R3u2dx-1/p+1?R3|u|p+1dx defined on H={u?W1,2(R3):?R3V(x)u2dx<?}.If ?<?1 in(0-1),we can easily prove that(0-1)has a least energy solution byNehari manifold method for any b>O,p ?(3,5)and(V1)holds,here ?1 is the first eigenvalue of the Schrodinger operator-?+V.In the case of ???1,however,this method is not suitable anymore.Throughout this paper,we obtain the existence of nontrivial solution,leastenergy solution,infinitely many nontrivial solutions to(0-1)based on mountain-pass theorem,Ekeland's variational principle,symmetric mountain-pass theorem and some skills in analysis.Firstly,we prove that for any b>O,there exists ?(b)>0 such that for any??[?1,?1+?(b)),(0-1)has a nontrivial solution u??H with I?(u?)>0 by using mountain-pass theorem.Moreever,for any sequence {?n}(?)[?1,?1+?(b))with ?n?n?1,there exists u?1?H with I'?1(u?1)=0 and I?1(u?1)>0,such that u?n?u?1 strongly in H.During the procedure,because of the existence of term?R3|?u|2dx which implies that(0-1)is not a pointwise identity,it is not easy to verify that the corresponding functional satisfies the P-S condition.We overcome those difficulties by studying the quality of P-S sequence in detail.Then we get the existence of least energy solution to(0-1)by studying a properminimazition problem and using Ekeland's variational principle.That is to say we prove that for any b>O,there exists ?1(b)?(0,?(b))such that for any??[?1,?1??(b)),(0-1)has a nonnegative least energy solution u??H with I?(u?)<0.Moreever,for any sequence {?n}(?)(?1,?1+?1(b))with ?n?n?1,u?n?u?1 strongly in H.Finally,with the help of a generalized version of symmetric mountain-pass theorem,we prove that(0-1)has infinitely many nontrivial solutions for any ??R.