Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $R^3$

Abstract

The paper considers the following nonhomogeneous Schrödinger-Maxwell system -Δu + u+λϕ(x) u =|u|^{p-1}u+g(x),\ x\in \mathbb{R}^3, -Δϕ= u^2, \ x\in \mathbb{R}^3, . \leqno{(SM)} where $\lambda >0$, $p\in (1,5)$ and $g(x)=g(|x|)\in L^2({\mathbb{R}}^3)\setminus 0$. There seems no any results on the existence of multiple solutions to problem (SM) for $p\in (1,3]$. In this paper, we find that there is a constant$C_p>0$ such that problem (SM) has at least two solutions for all $p\in (1,5)$ provided $\Vert g{\Vert }_{L^2}\le C_p$, but only for $p\in (1,2]$ we need $\lambda >0$ is small. Moreover, $C_p=\frac{(p-1)}{2p}[\frac{(p+1)S^{p+1}}{2p}{]}^{1/(p-1)}$, where $S$ is the Sobolev constant.

12 pages