Vol. 304, No. 1, 2020

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Pseudoindex theory and Nehari method for a fractional Choquard equation

Min Liu and Zhongwei Tang

Vol. 304 (2020), No. 1, 103–142
Abstract

We study the following nonlinear fractional Choquard equation:

ε2s(Δ)sw + V (x)w = εθW(x)[I θ (W|w|p)]|w|p2w,x N, ()

where ε > 0, s (0,1), N > 2s, Iθ is the Riesz potential with order θ (0,N), p [2, N+θ N2s), minNV > 0 and inf NW > 0. By specifying the ranges and interdependence of linear and nonlinear potentials, we achieve the existence, convergence, concentration, and decay estimate of positive groundstates for . The multiplicity of semiclassical solutions is established via pseudoindex theory. The existence of sign-changing solutions is constructed by minimizing the energy on Nehari nodal set.

Keywords
Fractional Choquard equation, pseudoindex, sign-changing solution, concentration
Mathematical Subject Classification 2010
Primary: 35J20
Secondary: 35R11
Milestones
Received: 9 October 2018
Revised: 11 August 2019
Accepted: 13 August 2019
Published: 18 January 2020
Authors
Min Liu
School of Mathematical Sciences
Beijing Normal University
Beijing
China
Zhongwei Tang
School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing
China