Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

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.