Electron. J. Differential Equations, Vol. 2024 (2024), No. 19, pp. 137.
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
$$
\epsilon^2 \Delta v +V(x)v
= \epsilon^{\alphaN}\Big(\int_{R^N}
\frac{v(y)^{2_\alpha^*}}{xy^{\alpha}}\,dy\Big) v^{2_\alpha^*2}v
+\thetav^{q2}v,\; x \in R^N,
$$
where \(\theta>0\), \(N\geq 3\),
\(0< \alpha<\min \{4,N1\},\max\{2,2^*1\}< q< 2^*\), \(2_\alpha^*= \frac{2N\alpha}{N2}\), \(V\) 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; signchanging 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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