# Stationary waves with prescribed $L^2$-norm for the planar Schr\"odinger-Poisson system

## TL;DR

This paper establishes existence and multiplicity of stationary wave solutions with fixed $L^2$-norm for a 2D Schr"odinger-Poisson system involving a logarithmic convolution potential, highlighting new mathematical challenges and solutions.

## Contribution

It introduces novel methods to find normalized solutions for the 2D Schr"odinger-Poisson system with a logarithmic kernel, expanding understanding beyond higher-dimensional cases.

## Key findings

- Proved existence of solutions under various parameter conditions.
- Established multiplicity results for normalized solutions.
- Developed new analytical techniques for the logarithmic convolution potential.

## Abstract

The paper deals with the existence of standing wave solutions for the Schr\"odinger-Poisson system with prescribed mass in dimension $N=2$. This leads to investigate the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namely $$ \left \{ \begin{aligned} - \Delta u & + \lambda u + \gamma \Bigl(\log {| \cdot |} * |u|^2 \Bigr) u =a |u|^{p-2} u \qquad \text{in $\mathbb R^2$,} \\ &\int_{\mathbb R^2} |u|^2 dx = c \end{aligned} \right. $$ where $c>0$ is a given real number. Under different assumptions on $\gamma \in \mathbb R$, $a \in \mathbb R$, $p>2$, we prove several existence and multiplicity results. Here $\lambda \in \mathbb R $ appears as a Lagrange parameter and is part of the unknowns. With respect to the related higher dimensional cases, the presence of the logarithmic kernel, which is unbounded from above and below, makes the structure of the solution set much richer and it forces the implementation of new ideas to catch the normalized solutions.