Localized nodal solutions for semiclassical quasilinear Choquard equations with subcritical growth

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 \(V\).