Liouville theorems and classification results for a nonlocal Schrödinger equation

Yutian Lei

doi:10.3934/dcds.2018236

Article Contents

2018, Volume 38, Issue 11: 5351-5377. Doi: 10.3934/dcds.2018236

This issue Previous Article Next Article Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

Liouville theorems and classification results for a nonlocal Schrödinger equation

Yutian Lei

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received: February 28, 2017

Revised: May 31, 2017

Published: August 2018

The research was supported by NSF of China (No. 11471164, 11671209), and PAPD of Jiangsu Higher Education Institutions

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In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation

$-Δ u = pu^{p-1}(|x|^{2-n}*u^p),\;\; u>0 \;\;in\;\; R^n, $

where $n ≥ 3$ and $p≥ 1$. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when $1 ≤ p <\frac{n+2}{n-2}$ by means of the method of moving planes to the following system

$\left\{ \begin{array}{l} - \Delta u = \sqrt p {u^{p - 1}}v,\;\;u > 0\;\;in\;\;{R^n},\\ - \Delta v = \sqrt p {u^p},\;\;v > 0\;\;in\;\;{R^n}.\end{array} \right.$

When $p = \frac{n+2}{n-2}$, all the positive solutions can be classified as

$u(x) = c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$

with the help of an integral system involving the Newton potential, where $c, t$ are positive constants, and $x^* ∈ R^n$. In addition, we also give other equivalent conditions to classify those positive solutions. When $p>\frac{n+2}{n-2}$, by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate $\frac{2}{p-1}$. Finally, we point out that the equation has positive stable solutions if and only if $p ≥ 1+\frac{4}{n-4-2\sqrt{n-1}}$.

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Mathematics Subject Classification: 35J10, 35J47, 35Q55, 45E10, 45G050.

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