Multiple solutions for a nonhomogeneous Schr\"odinger-Maxwell system in $R^3$

TL;DR

This paper proves the existence of multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in three-dimensional space for certain parameter ranges, extending previous results to cases with $p$ in (1,5).

Contribution

It establishes the existence of at least two solutions for the Schrödinger-Maxwell system when the nonhomogeneous term is sufficiently small, covering the full range of $p$ in (1,5), which was previously unresolved.

Findings

Multiple solutions exist for all $p$ in (1,5) when $ orm{g}_{L^2}$ is small.

For $p$ in (1,2], small $ heta$ ensures solutions exist.

Explicit bounds for $ orm{g}_{L^2}$ are provided based on $p$ and Sobolev constants.

Abstract

The paper considers the following nonhomogeneous Schr\"odinger-Maxwell system -\Delta u + u+\lambda\phi (x) u =|u|^{p-1}u+g(x),\ x\in \mathbb{R}^3, -\Delta\phi = 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 $\|g\|_{L^2} \leq C_p$, 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.