Multiple positive solutions for Schrodinger-Poisson systems involving critical nonlocal term

Abstract

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term $$\{\begin{array}{llc}-\Delta u+u-K(x)\varphi |u{|}^{3}u=\lambda f(x)|u{|}^{q-2}u,& x\in {\mathbb{R}}^3,-\Delta \varphi =K(x)|u{|}^5,& x\in {\mathbb{R}}^3,\end{array}$$ where $1<q<2$ and $\lambda >0$ is a parameter. Under suitable assumptions on $K(x)$ and $f(x)$, there exists ${\lambda }_0={\lambda }_0(q,S,f,K)>0$ such that for any $\lambda \in (0,{\lambda }_0)$, the above Schrödinger-Poisson system possesses at least two positive solutions by standard variational method, where a positive least energy solution will also be obtained.

arXiv admin note: substantial text overlap with arXiv:1702.03785

References in corpus (1)

• Multiple positive solutions for a class of Kirchhoff type problems involving general critical growth