Sign-changing solutions for a fractional Choquard equation with power nonlinearity. (August 2022)

• Record Type: Journal Article
• Authors: Zhao, Shunneng; Yu, Yuanyang
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: 35A15 -- 35J60; Fractional Choquard equation -- Sign-changing solutions -- Variational methods; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112917
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study the following fractional Choquard equation (FC) ( − Δ ) s u + V ( x ) u = ( I α ∗ | u | p ) | u | p − 2 u, in R N, where s ∈ ( 0, 1 ), N ≥ 3, V ( x ) is continuous potential function, I α : R N → R is the Riesz potential of order α defined by I α ( x ) = | x | α − N for every x ∈ R N ∖ { 0 }, α ∈ ( 0, N ), ∗ denotes the convolution operator, 2 < p < N + α N − 2 s = 2 α, s ∗, 2 α, s ∗ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and fractional Laplace operator and the operator ( − Δ ) s stands for the fractional Laplacian of order s . Combining constraint variational method, quantitative deformation lemma and the Brouwer degree theory, we prove that (FC) possesses one least energy sign-changing solution u 0 . Moreover, we show that the energy of u 0 is strictly larger than 2 p − 2 p − 1 times and strictly less than four times the ground state energy.