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

In this paper, we study the following Schr"{o}dinger-Born-infeld system with a general nonlinearity $$\{\begin{array}{ll}-\mathrm{△}u+u+\varphi u=f(u)+\mu \mathrm{\mid }u{\mathrm{\mid }}^{4}u& in{\mathbb{R}}^3,\\ -\text{div}{\displaystyle \left(\frac{\mathrm{\nabla }\varphi }{\sqrt{1-\mathrm{\mid }\mathrm{\nabla }\varphi {\mathrm{\mid }}^2}}\right)=u^2}& in{\mathbb{R}}^3,\\ u(x)\to 0,\varphi (x)\to 0,& \text{as}x\to \mathrm{\infty },\end{array}$$ 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 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 $f(u)=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ for $p\in (2,5\mathrm{/}2]$ and $\mu =0$ which was left in \cite{Azzollini19} as an open problem.