Normalized solutions of quasilinear Schr\"odinger-Poisson system with critical nonlinear term in bounded domain

TL;DR

This paper proves the existence of multiple normalized solutions for a quasilinear Schr"odinger-Poisson system with critical nonlinearity in a bounded domain, using truncation, genus theory, and concentration-compactness methods.

Contribution

It introduces a novel approach combining truncation and genus theory to find multiple solutions in a critical nonlinear Schr"odinger-Poisson system.

Findings

Established existence of multiple solutions

Applied concentration-compactness to handle critical exponent

Derived asymptotic reduction to classical Schr"odinger-Poisson system

Abstract

This work examines a quasilinear Schr\"odinger-Poisson system involving a critical nonlinearity, expressed as \[ -\Delta u + \phi u + \lambda u = |u|^{q-2} u + |u|^4 u, \quad x \in \Omega_r, \] \[ -\Delta \phi - \varepsilon^4 \Delta_4 \phi = u^2, \qquad\qquad\qquad\quad\ x \in \Omega_r, \] \[ \enspace u = \phi = 0, \qquad\qquad\qquad\qquad\qquad\enspace\ \ \,x \in \partial \Omega_r \] subject to the normalized condition \[ \int_{\Omega_r} |u|^2\, \mathrm d x = b^2. \] Here $\varepsilon > 0$, $q \in (2, 8/3)$, $\Omega_r \subset \mathbb R^3$ is a bounded domain. By means of a truncation method combined with genus theory, we establish the existence of multiple families of normalized solutions. Due to the presence of a critical exponent in the nonlinear term, the associated energy functional fails to satisfy the usual compactness properties. To address this issue, we invoke the concentration-compactness principle. Furthermore, we derive the asymptotic result that the aforementioned system reduces to the classical Schr\"odinger-Poisson system (with $\varepsilon = 0$). Our findings extend several recent results concerning problems of this type.