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

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 $\mathbb{R}^{2}$.