On the Schr\"odinger-Maxwell system involving sublinear terms

In this paper we study the coupled Schr"odinger-Maxwell system $$\{\begin{array}{lll}-\mathrm{△}u+u+\varphi u=\lambda \alpha (x)f(u)& \mathrm{i}\mathrm{n}& {\mathbb{R}}^3,\\ -\mathrm{△}\varphi =u^2& \mathrm{i}\mathrm{n}& {\mathbb{R}}^3,\end{array}$$ where $\alpha\in L^\infty(\mathbb R^3)\cap L^{6/(5-q)}(\mathbb R^3)$forsome$q\in (0,1)$,andthecontinuousfunction$f:$\mathbb{R}$\to$\mathbb{R}$$ is superlinear at zero and sublinear at infinity, e.g., $f(s)=\min (\mathrm{\mid }s{\mathrm{\mid }}^r,\mathrm{\mid }s{\mathrm{\mid }}^p)$ with $0<r<1<p\mathrm{.}$ Depending on the range of $\lambda >0$, non-existence and multiplicity results are obtained.