Quasilinear Schr\"odinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions

TL;DR

This paper investigates the existence and asymptotic behavior of solutions to a quasilinear Schr"odinger-Poisson system with exponential critical nonlinearity in a bounded domain, analyzing the limit as a parameter approaches zero.

Contribution

It establishes the existence of solutions for the system with critical exponential growth and describes their convergence to solutions of the limiting system as the parameter tends to zero.

Findings

Existence of nontrivial solutions for the system with exponential critical nonlinearity.

Convergence of solutions to the limiting Schr"odinger-Poisson system as the parameter approaches zero.

Analysis of the system under exponential critical growth conditions.

Abstract

In this paper we consider the following quasilinear Schr\"odinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi - \varepsilon^{4}\Delta_4 \phi = u^{2} & \ \mbox{in } \Omega, u=\phi=0 & \ \mbox{on } \partial\Omega \end{array} \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\varepsilon}, \phi_{\varepsilon})$ and then we show that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial solution of the associated Schr\"odinger-Poisson system, that is by making $\varepsilon=0$ in the system above.