Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 11, pp. 1-29.

Title: Localized nodal solutions for  semiclassical quasilinear Choquard equations with subcritical growth
       
Authors: Bo Zhang (Yunnan Normal Univ., Kunming, Yunnan, China)
         Xiangqing Liu (Yunnan Normal Univ., Kunming, Yunnan, China)

Abstract:    
In this article,  we study the existence of localized nodal solutions for semiclassical
quasilinear Choquard equations with subcritical growth
$$
-\varepsilon^p \Delta_{p} v +V(x)|v|^{p-2}v
= \varepsilon^{\alpha-N} |v|^{q-2}v
\int_{\mathbb{R}^N}
\frac{|v(y)|^q}{|x-y|^{\alpha}}\,dy,\quad x \in \mathbb{R}^N\,,
$$
where $N\geq 3$, $1<p<N$, $0<\alpha<\min\{2p,N-1\}$, $p<q<p_\alpha^*$,
$p_\alpha^*= \frac{p(2N-\alpha)}{2(N-p)}$, V is a bounded function.
By the perturbation method and the method of invariant sets of descending flow,
for small $\varepsilon$ we establish the existence of a sequence of localized nodal
solutions concentrating near a given local  minimum point of the potential function V.

Submitted September 6, 2021. Published February 10, 2022.
Math Subject Classifications: 35B20, 35B40.
Key Words: Quasilinear Choquard equation; nodal solutions; perturbation method.