On planar Schrodinger-Poisson systems with repulsive interactions in the mass supercritical regime

TL;DR

This paper studies solutions with fixed mass for a planar Schrödinger-Poisson system with repulsive interactions, proving existence of solutions and analyzing their behavior as the domain expands to the whole plane.

Contribution

It establishes the existence of ground and high-energy solutions for the system in large domains and analyzes their asymptotic behavior, addressing open problems in the field.

Findings

Energy functional is unbounded on the Pohozaev manifold.

Existence of ground state and high-energy solutions in large domains.

Solutions' behavior as domain extends to the entire plane.

Abstract

In this paper, we investigate solutions with prescribed $L^{2}$-norm (i.e., prescribed mass) for the planar Schr\"{o}dinger-Poisson (SP) equation% \begin{equation*} -\Delta u+\lambda u+\alpha \left( \log |\cdot |\ast |u|^{2}\right) u=|u|^{p-2}u,\ \text{in}\ \Omega_{R} , \end{equation*}% where $\lambda \in \mathbb{R}$ is unknown, $\alpha <0,p>4$ and $\Omega_{R} \subseteq \mathbb{R}^{2}$ is a domain. First, we prove that the energy functional $J$ corresponding to the SP equation in $\mathbb{R}^{2}$ is unbounded both above and below on the Pohozaev manifold $\mathcal{P}$; this explains the reason why the minimax level of $J$ is difficult to determine, as referenced in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Second, we establish the existence of a ground state and a high-energy solution, both with positive energy in a large bounded domain $\Omega_{R} $, which is a substantial advancement in addressing an open problem proposed in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Finally, we analyze the asymptotic behavior of solutions as the domain $\Omega_{R} $ is extended to the entire space $\mathbb{R}^{2}$.