| In this paper,we study the following nonlinear Kirchhoff type elliptic equation-(a+b∫R3|▽u|2dx)△u+V(x)u=μu+|u|p-1u,x∈R3,u∈H1R3),(0-1)where a,b>0 are constants,μ>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 Iμ(u)=1/2∫R3a|▽u|2+V(x)u2dx+b/4(∫R3|▽u|2dx)2-μ/2∫R3u2dx-1/p+1∫R3|u|p+1dx defined on H={u∈W1,2(R3):∫R3V(x)u2dx<∞}.If μ<μ1 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 μ1 is the first eigenvalue of the Schrodinger operator-△+V.In the case of μ≥μ1,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 δ(b)>0 such that for anyμ∈[μ1,μ1+δ(b)),(0-1)has a nontrivial solution uμ∈H with Iμ(uμ)>0 by using mountain-pass theorem.Moreever,for any sequence {μn}(?)[μ1,μ1+δ(b))with μn→nμ1,there exists uμ1∈H with I’μ1(uμ1)=0 and Iμ1(uμ1)>0,such that uμn→uμ1 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 δ1(b)∫(0,δ(b))such that for anyμ∈[μ1,μ1＋δ(b)),(0-1)has a nonnegative least energy solution uμ∈H with Iμ(uμ)<0.Moreever,for any sequence {μn}(?)(μ1,μ1+δ1(b))with μn→nμ1,uμn→uμ1 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 μ∈R. |
