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

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 $λ>0$, $p\in(1,5)$ 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 $\|g\|_{L^2} \leq C_p$, but only for $p\in(1,2]$ we need $λ>0$ 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.

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