#### 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:

 ${\epsilon }^{2s}{\left(-\Delta \right)}^{s}w+V\left(x\right)w={\epsilon }^{-\theta }W\left(x\right)\left[{I}_{\theta }\ast \left(W|w{|}^{p}\right)\right]|w{|}^{p-2}w,\phantom{\rule{1em}{0ex}}x\in {ℝ}^{N},$ ($\star$)

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
Primary: 35J20
Secondary: 35R11
##### Milestones
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