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# 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

#### 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.

D. Cao was supported by NNSF of China (No. 11831009) and Chinese Academy of Sciences (No. QYZDJ-SSW-SYS021).

W. Dai was supported by the NNSF of China (No. 11971049), the Fundamental Research Funds for the Central Universities and the State Scholarship Fund of China (No. 201806025011).

Received 24 August 2020

Published 10 May 2021