Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 19, pp. 1-37.

Title: Localized nodal solutions for semiclassical Choquard equations with critical growth

Authors: Bo Zhang (Sichuan Univ. of Arts and Science, Dazhou, China)
         Wei Zhang (Yunnan Univ. of Finance and Economics, Kunming, China)

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.