Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 2

Existence and symmetry of solutions to 2-D Schrödinger–Newton equations

Pages: 113 – 156

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n2.a3

Authors

Daomin Cao (School of Mathematics and Information Science, Guangzhou University, Guangzhou, China)

Wei Dai (School of Mathematical Sciences, Beihang University (BUAA), Beijing, China; and Institut Galilée, LAGA, UMR 7539, Université Sorbonne Paris Cité, Villetaneuse, France)

Yang Zhang (School of Mathematics and Statistics, Central South University, Changsha, China)

Abstract

In this paper, we consider the following 2-D Schrödinger–Newton equations\[-\Delta u + a(x)u + \frac{\gamma}{2\pi} (\operatorname{log}(\lvert \cdot \rvert) \ast {\lvert u \rvert}^p) {\lvert u \rvert}^{p-2} u = b {\lvert u \rvert}^{q-2} u \quad \textrm{in} \; \mathbb{R}^2 \; \textrm{,}\]where $a \in C(\mathbb{R}^2)$ is a $\mathbb{Z}^2$-periodic function with $\operatorname{inf}_{\mathbb{R}^2} a \gt 0, \gamma \gt 0, b \geq 0, p \geq 2$ and $q \geq 2$. By using ideas from [14, 22, 43], under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for $p \geq 2$ and $q \geq 2p-2$ via variational methods. The auxiliary functional $J_1$ plays a key role in the case $p \geq 3$. We also prove the radial symmetry of positive solutions (up to translations) for $p \geq 2$ and $q \geq 2$. The corresponding results for planar Schrödinger–Poisson systems will also be obtained. Our theorems extend the results in [14] from $p=2$ to general $p \geq 2$ and the results in [22] from $p=2$ and $b=1$ to general $p \geq 2$ and $b \geq 0$.

Keywords

logarithmic convolution potential, variational methods, Schrödinger–Newton equations, Schrödinger–Poisson systems, positive solutions, radial symmetry

2010 Mathematics Subject Classification

Primary 35J20. Secondary 35B06, 35B09, 35Q40.

Received 24 August 2020

Published 10 May 2021