A $p$-Laplacian Schr\"odinger--Maxwell system with rapidly growing nonlinearities

Abstract

Let $\Omega\subset\R^N$, $N\ge2$, be a bounded domain and let $1<p<\infty$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schr\"odinger--Maxwell system \[ \begin{cases} -\Delta_p u+\psi G'(u)=f&\text{in }\Omega,\\ -\Delta_p\psi=G(u)&\text{in }\Omega,\\ u=\psi=0&\text{on }\partial\Omega, \end{cases} \] where $G\in C^1(\R)$ is even, convex, nonnegative and nontrivial, with $G(0)=0$, and satisfies \[ G(t)\lesssim 1+|G'(t)| \qquad\text{for large }|t|. \] Under the assumption \[ |f|\ln(1+|f|)\in L^1(\Omega), \] we prove the existence of a finite-energy weak solution and show that the solution is a saddle point of the associated functional.