Pseudoindex theory and Nehari method for a fractional Choquard equation

Min Liu and Zhongwei Tang

Vol. 304 (2020), No. 1, 103–142

DOI: 10.2140/pjm.2020.304.103

Abstract

We study the following nonlinear fractional Choquard equation:

$${\epsilon }^{2s}{\left(-\Delta \right)}^{s}w+V\left(x\right)w={\epsilon }^{-\theta }W\left(x\right)\left[I_{\theta }\ast \left(W|w{|}^p\right)\right]|w{|}^{p-2}w,x\in {ℝ}^N,$$

($\star$)

where $\epsilon >0$, $s\in \left(0,1\right)$, $N>2s$, $I_{\theta }$ is the Riesz potential with order $\theta \in \left(0,N\right)$, $p\in \left[2,\frac{N+\theta }{N-2s}\right)$, $\underset{{ℝ}^N}{min}V>0$ and $\underset{{ℝ}^N}{inf}W>0$. By specifying the ranges and interdependence of linear and nonlinear potentials, we achieve the existence, convergence, concentration, and decay estimate of positive groundstates for $\star$. 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