A perturbation approach for the Schr\"odinger-Born-Infeld system: solutions in the subcritical and critical case

TL;DR

This paper introduces a new perturbation method to establish the existence and multiplicity of solutions for a coupled Schrödinger-Born-Infeld system, covering subcritical and critical nonlinearities, including previously unresolved cases.

Contribution

The paper develops a novel perturbation approach to solve the Schrödinger-Born-Infeld system, extending results to critical cases and specific nonlinearities previously left open.

Findings

Proved existence of solutions in subcritical and critical cases.

Established multiplicity of solutions under certain conditions.

Covered the case f(u)=|u|^{p-1}u for p in (2, 5/2] with μ=0.

Abstract

In this paper, we study the following Schr\"{o}dinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+\phi u=f(u)+\mu|u|^4u\,\,&\mbox{in}\,\,\R^3,\\ -\textrm{div}\displaystyle\bigg(\frac{\nabla\phi}{\sqrt{1-|\nabla\phi|^2}}\bigg)=u^2&\mbox{in}\,\,\R^3,\\ u(x)\rightarrow0,\,\,\phi(x)\rightarrow0,&\,\text{as}\,\,x\rightarrow\infty, \end{array} \right. $$ where $\mu\geq0$ and $f\in C(\R,\R)$ satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schr\"{o}dinger equation and the Born-Infeld theory. We use a new perturbation approach to prove the existence and multiplicity of nontrivial solutions of the above system in the subcritical and critical case. We emphasise that our results cover the case $f(u)=|u|^{p-1}u$ for $p\in(2,{5}/{2}]$ and $\mu=0$ which was left in \cite{Azzollini19} as an open problem.