  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.
