A perturbation approach for the Schrödinger-Born-Infeld system: solutions in the subcritical and critical case

Abstract

In this paper, we study the following Schrödinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+ϕu=f(u)+μ|u|^4u\,\,&\mbox{in}\,\,\R^3,\\ -\textrm{div}\displaystyle\bigg(\frac{\nablaϕ}{\sqrt{1-|\nablaϕ|^2}}\bigg)=u^2&\mbox{in}\,\,\R^3,\\ u(x)\rightarrow0,\,\,ϕ(x)\rightarrow0,&\,\text{as}\,\,x\rightarrow\infty, \end{array} \right. $$ where $\mu \ge 0$ and $f\in C(\mathbb{R},\mathbb{R})$ satisfies suitable assumptions. This system arises from a suitable coupling of the nonlinear Schrödinger 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.

References in corpus (2)

• On the electrostatic Born-Infeld equation with extended charges
• A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity