On the Schrödinger-Maxwell system involving sublinear terms

In this paper we study the coupled Schrödinger-Maxwell system $\left\{ \begin{array}{lll} -\triangle u+u +ϕu=λα(x) f(u)& {\rm in} & \mathbb R^3,\\ -\triangle ϕ=u^2 & {\rm in} & \mathbb R^3, \end{array}\right.$ where $α\in L^\infty(\mathbb R^3)\cap L^{6/(5-q)}(\mathbb R^3)$ for some $q\in (0,1)$, and the continuous function $f:\mathbb{R}\to\mathbb{R}$ is superlinear at zero and sublinear at infinity, e.g., $f(s) = \min(|s|^r,|s|^p)$ with $0<r<1<p.$ Depending on the range of $λ>0$, non-existence and multiplicity results are obtained.

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