# American Institute of Mathematical Sciences

March  2020, 19(3): 1351-1365. doi: 10.3934/cpaa.2020066

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

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

* Corresponding author

Received  March 2019 Revised  August 2019 Published  November 2019

Fund Project: 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).

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
 $\lambda$
.
Citation: Gui-Dong Li, Yong-Yong Li, Xiao-Qi Liu, Chun-Lei Tang. A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066
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