Localized nodal solutions for semiclassical Choquard equations with critical growth

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 \(V\). For more information see https://ejde.math.txstate.edu/Volumes/2024/19/abstr.html