Existence of normalized solutions for the planar Schr\"odinger-Poisson system with exponential critical nonlinearity

TL;DR

This paper proves the existence of normalized solutions for a planar Schr"odinger-Poisson system with exponential critical nonlinearity, extending previous results and addressing the challenges posed by the critical growth in two dimensions.

Contribution

It establishes new existence results for solutions with prescribed mass in a Schr"odinger-Poisson system involving exponential critical growth, with the Lagrange multiplier as an unknown.

Findings

Existence of solutions with prescribed $L^2$ norm.

Extension of previous results to exponential critical growth.

Inclusion of the Lagrange multiplier as an unknown parameter.

Abstract

In the present work we are concerned with the existence of normalized solutions to the following Schr\"odinger-Poisson System $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ } \mathbb{R}^2 , \\ \intR |u(x)|^2 dx = c,\ c> 0 , \end{array} \right. $$ for $\mu \in \R $ and a nonlinearity $f$ with exponential critical growth. Here $ \lambda\in \R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in \cite{Ji} and \cite{[cjj]}.