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

TL;DR

This paper investigates the Schr"odinger-Maxwell system with sublinear nonlinearities, establishing conditions for the existence, non-existence, and multiplicity of solutions based on the parameter range.

Contribution

It introduces new existence and multiplicity results for the Schr"odinger-Maxwell system with sublinear terms, extending previous analyses to more general nonlinearities.

Findings

Existence of solutions for certain parameter ranges

Non-existence results for other parameter ranges

Multiple solutions under specific conditions

Abstract

In this paper we study the coupled Schr\"odinger-Maxwell system $$\left\{ \begin{array}{lll} -\triangle u+u +\phi u=\lambda \alpha(x) f(u)& {\rm in} & \mathbb R^3,\\ -\triangle \phi =u^2 & {\rm in} & \mathbb R^3, \end{array}\right. $$ where $\alpha\in L^\infty(\mathbb R^3)\cap L^{6/(5-q)}(\mathbb R^3)$ 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 $\lambda>0$, non-existence and multiplicity results are obtained.