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March  2020, 19(3): 1563-1579. doi: 10.3934/cpaa.2020078

## Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent

 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author

Received  June 2019 Revised  August 2019 Published  November 2019

Fund Project: X. H. Tang was partially supported by the National Natural Science Foundation of China (No: 11571370).

This paper is dedicated to studying the Choquard equation
 $\begin{equation*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = (I_{\alpha}\ast|u|^{p})|u|^{p-2}u+g(u),\; \; \; \; \; x\in\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}) ,\\ \end{array} \right. \end{equation*}$
where
 $N\geq4$
,
 $\alpha\in(0, N)$
,
 $V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R})$
is sign-changing and periodic,
 $I_{\alpha}$
is the Riesz potential,
 $p = \frac{N+\alpha}{N-2}$
and
 $g\in\mathcal{C}(\mathbb{R}, \mathbb{R})$
. The equation is strongly indefinite, i.e., the operator
 $-\Delta+V$
has infinite-dimensional negative and positive spaces. Moreover, the exponent
 $p = \frac{N+\alpha}{N-2}$
is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on
 $g$
, we obtain the existence of nontrivial solutions for this equation.
Citation: Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078
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