A $p$-Laplacian Schrödinger--Maxwell system with rapidly growing nonlinearities

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$, be a bounded domain and let $1<p<\mathrm{\infty }$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schrödinger–Maxwell system $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u+\psi G^{\mathrm{\prime }}(u)=f& \text{in }\mathrm{\Omega },\\ -{\mathrm{\Delta }}_p\psi =G(u)& \text{in }\mathrm{\Omega },\\ u=\psi =0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $G\in C^1(\mathbb{R})$ is even, convex, nonnegative and nontrivial, with $G(0)=0$, and satisfies $$G(t)\lesssim 1+\mathrm{\mid }{G}^{\mathrm{\prime }}(t)\mathrm{\mid }\text{for large }\mathrm{\mid }t\mathrm{\mid }\mathrm{.}$$ Under the assumption $$\mathrm{\mid }f\mathrm{\mid }\ln (1+\mathrm{\mid }f\mathrm{\mid })\in L^1(\mathrm{\Omega }),$$ we prove the existence of a finite-energy weak solution and show that the solution is a saddle point of the associated functional.