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

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.