ON A CLASS OF CHOQUARD-TYPE EQUATION WITH UPPER CRITICAL EXPONENT AND INDEFINITE LINEAR PART

Huiling Wu; Haiping Xu

doi:10.11948/20210024

2022 Volume 12 Issue 2

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Huiling Wu, Haiping Xu. ON A CLASS OF CHOQUARD-TYPE EQUATION WITH UPPER CRITICAL EXPONENT AND INDEFINITE LINEAR PART[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 464-478. doi: 10.11948/20210024

Citation:

Huiling Wu, Haiping Xu. ON A CLASS OF CHOQUARD-TYPE EQUATION WITH UPPER CRITICAL EXPONENT AND INDEFINITE LINEAR PART[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 464-478. doi: 10.11948/20210024

ON A CLASS OF CHOQUARD-TYPE EQUATION WITH UPPER CRITICAL EXPONENT AND INDEFINITE LINEAR PART

Huiling Wu^1, ,,
Haiping Xu¹

1.
College of Mathematics and Data Science (Software College), Minjiang University, Fuzhou, 350108, China

Corresponding author: Email: huilingwu@mju.edu.cn(H. Wu)
Fund Project: This work is partially supported by the Natural Science Foundation of Fujian Province (No. 2021J05206) and the Scientific Research Foundation of Minjiang University (No. mjy18014)

Abstract

Abstract

The existence of ground states to the strongly indefinite Choquard type equation

$ \left\{\begin{array}{lcl} -\Delta u+(V(x)-W(x))u=a(x)(I_{\alpha}*|u|^{\frac{2_{\alpha}^{*}}{2}})|u|^{\frac{2_{\alpha}^{*}}{2}-2}u+b(x)f(u), \ x\in {{ \mathbb{R}}}^{N}, \\ u\in H^{1}({{ \mathbb{R}}}^{N}), \end{array}\right. $

is proved, where $N\ge3$, $\alpha\in (0, N)$, $ 2_{\alpha}^{*}=\frac{2(N+\alpha)}{N-2},$ $I_{\alpha}$ denotes the Riesz potential, and $zhongwenzy$ belongs to the gap of the spectrum of $-\Delta+V$. We consider the asymptotically periodic case, i.e., $V$ is periodic in $x$, $\lim_{|x|\to+\infty}W(x)=0, $ and $a, \, b$ are asymptotically periodic in $x$. Some results in the literature are completed.
- Critical exponent /
- Choquard equation /
- ground state /
- strongly indefinite functional
MSC: 35J20, 35Q55

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