Localized nodal solutions for semiclassical Choquard equations

Abstract

In this paper, we study the existence of localized nodal solutions for the semiclassical Choquard equation −ε2Δu+V(x)u=ε−α(Iα*|u|p)|u|p−2u for x∈RN. We establish for small ɛ the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function V by using the perturbation method and the method of invariant sets of descending flow.

Localized nodal solutions for semiclassical Choquard equations with critical growth

“…where N ≥ 3, 0 < α < min{4, N − 1}, the potential function V satisfies (A1) and (A2). Zhang and Liu [45] investigated the semiclassical quasi-linear Choquard equation with subcritical growth, and obtained a conclusion similar to that of [17] in 2022.…”

“…More results for the semiclassical Choquard equation with subcritical growth we refer [8,17,18,27,37,44,45,48] and references therein.…”

“…In recent years, there are have been many results of localized nodal solution for the Choquard equation. In 2021, He and Liu [17] proved existence and concentration of infinitely many sign-changing solutions for the following Choquard equation…”

“…where 2 < 2β < r. Note that the method of invariant sets of descending flow [28] can not fit well for the functional Γ ε,ν , so we also use the perturbation method [17] to overcome this difficulty. For t ∈ R + , we define…”

Concentration of nodal solutions for semiclassical quadratic Choquard equations

“…For the existence and qualitative properties of solutions for the nonlinear Choquard equation (1.1), we refer the reader to [2,3,5,9,10,14,17,26,27,31,32,34,35] and references therein. In particular, for p > 2, the existence of nodal solutions for the Choquard equation is an appealing aspect which is investigated in [5,8,11,13,15,16,23] by the variational method. In the physical case, for p = 2, the existence of nodal solutions for (1.1) only has few results.…”