Electron. J. Differential Equations, Vol. 2024 (2024), No. 19, pp. 1-37.

Localized nodal solutions for semiclassical Choquard equations with critical growth

Bo Zhang, Wei Zhang

In this article, we study the existence of localized nodal solutions for semiclassical Choquard equation with critical growth $$ -\epsilon^2 \Delta v +V(x)v = \epsilon^{\alpha-N}\Big(\int_{R^N} \frac{|v(y)|^{2_\alpha^*}}{|x-y|^{\alpha}}\,dy\Big) |v|^{2_\alpha^*-2}v +\theta|v|^{q-2}v,\; x \in R^N, $$ where \(\theta>0\), \(N\geq 3\), \(0< \alpha<\min \{4,N-1\},\max\{2,2^*-1\}< q< 2^*\), \(2_\alpha^*= \frac{2N-\alpha}{N-2}\), \(V\) is a bounded function. By the perturbation method and the method of invariant sets of descending flow, we establish for small \(\epsilon\) the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function \(V\).

Submitted October 17, 2023. Published February 16, 2024.
Math Subject Classifications: 35B20, 35Q40.
Key Words: Choquard equation; sign-changing solutions; nodal solutions; variational perturbation method; semiclassical states.
DOI: 10.58997/ejde.2023.19

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Bo Zhang
School of Mathematics
Sichuan University of Arts and Science
Dazhou 635000, China
email: zhangbo371013@163.com
Wei Zhang
School of Statistics and Mathematics
Yunnan University of Finance and Economics
Kunming 650221, China
email: weizyn@163.com

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