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

Source abstract

The paper deals with the existence of standing wave solutions for the Schrödinger--Poisson system with prescribed mass in dimension $N=2$. This leads to investigating the existence of normalized solutions for an integrodifferential equation involving a logarithmic convolution potential, namely, $- \Delta u + \lambda u + \gamma \bigl(\log {| \cdot |} * |u|^2 \bigr) u =a |u|^{p-2} u$ in ${\mathbb R}^2$, $\int_{{\mathbb R}^2} |u|^2 dx = c$, 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, forcing the implementation of new ideas to catch the normalized solutions.