A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities

Gui-Dong Li; Yong-Yong Li; Xiao-Qi Liu; Chun-Lei Tang

doi:10.3934/cpaa.2020066

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2020, Volume 19, Issue 3: 1351-1365. Doi: 10.3934/cpaa.2020066

This issue Previous Article Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation Next Article Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold

A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^* Corresponding author
^* Corresponding author

Received: February 28, 2019

Revised: July 31, 2019

Published: February 2020

This work is partially supported by National Natural Science Foundation of China (No.11971393) and Graduate Student Scientific Research Innovation Projects in Chongqing (No. CYB19082)

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Abstract

In this paper, we investigate the following Choquard equation

$ \begin{equation*} -\Delta u+V(x)u = \lambda(I_\alpha*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $

where $ N\geq 3, \lambda>0, \alpha\in (0, N) $, $ V $ is an asymptotically periodic potential, $ I_\alpha $ is the Riesz potential, the nonlinearity term $ F(s) = \int_{0}^{s}f(t)dt $ and $ f $ is only locally defined in a neighborhood of $ u = 0 $ and satisfies the suitable conditions. By using the Nehari manifold and the Moser iteration, we prove the existence of positive solutions for the equation with sufficiently large $ \lambda $.

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Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35J50, 35D30.

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