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

Abstract:
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.2024.19

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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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