一类带有两个参数的临界薛定谔-泊松方程的多重解

Multiplicity of Solutions for a Class of Critical Schrodinger-Poisson System with Two Parameters

该文研究如下一类临界薛定谔-泊松方程{-△u+λV(x)u+Фu=μu|^(p-2)u+|u|^(4)u,x∈R^(3)-△Ф=u^(2),x∈R^(3),其中λ>0,μ>0是两个参数,p∈(4,6),V满足一些势井条件.当参数λ充分大时,利用变分法证明了基态解的存在性:以及随着λ→∞时,这些解的渐近行为.另外,在参数λ充分大和μ充分小时,利用Ljusternik-Schnirelmann理论,到了多重解的存在性定理.

In this paper,we consider the following critical Schrodinger-Poisson system{-△u+λV(x)u+Фu=μu|^(p-2)u+|u|^(4)u,x∈R^(3)-△Ф=u^(2),x∈R^(3),whereλ,μare two positive parameters,p∈(4,6)and V satisfies some potential well conditions.By using the variational arguments,we prove the existence of ground state solutions forλlarge enough andμ>0,and their asymptotical behavior asλ→∞.Moreover,by using LusternikSchnirelmann theory,we obtain the existence of multiple solutions ifλis large andμis small.