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

Abstract

Let $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, be a bounded domain and let $1<p<\infty$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schrödinger--Maxwell system \[ \begin{cases} -Δ_p u+ψG'(u)=f &\text{in }Ω,\\ -Δ_pψ=G(u) &\text{in }Ω,\\ u=ψ=0 &\text{on }\partialΩ, \end{cases} \] where $G\in C^1(\mathbb{R})$ 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 \[ |f|\ln(1+|f|)\in L^1(Ω), \] we prove the existence of a finite-energy weak solution and show that the solution is a saddle point of the associated functional.