分数阶Schrodinger-Poisson-Slater方程基态解的存在性

Existence of Ground State for a Fractional Schrodinger-Poisson-Slater System

本文研究下面的分数阶Schrodinger-Poisson-Slater系统{(-△)^(s)u+V(x)u+λФ(x)u=μu+|u|^(p-1)u,x∈R^(3),(-△)^(s)Ф=u^(2),lim|x|→+∞Ф(x)=0,其中s∈(1/2,1),p∈(1,2),μ∈R,λ>0,V∈C(R^(N),R^(+))以及lim_(|x|→+∞)V(x)=∞.我们应用变分法证明了当参数λ,μ取值在适当的范围时,上述问题存在基态解.进一步,我们还研究了这些基态解在λ→0情况下的渐近行为.

We study the following fractional Schrodinger-Poisson-Slater system{(-△)^(s)u+V(x)u+λФ(x)u=μu+|u|^(p-1)u,x∈R^(3),(-△)^(s)Ф=u^(2),lim|x|→+∞Ф(x)=0,where s∈(1/2,1),p∈(1,2),μ∈R,λ>0,V∈C(R^(N),R^(+))and lim_(|x|→+∞)V(x)=∞.By using variational methods we show that the above problem has a ground state under suitable assumptions on the parametersλandμ.Moreover,the asymptotical behavior of ground states asλ→O has also been discussed.