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

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