On fractional Schrodinger systems of Choquard type

TL;DR

This paper proves existence and stability of standing waves for fractional Schrödinger-Choquard equations and systems using concentration compactness, extending results to multiple nonlinearities and general convolution potentials.

Contribution

It introduces new methods to establish existence and stability of standing waves for fractional Schrödinger-Choquard systems, including multiple nonlinearities and general convolution potentials.

Findings

Proved existence of standing waves using concentration compactness.

Established stability results for these standing waves.

Extended methods to systems with multiple nonlinearities and general potentials.

Abstract

In this article, we first employ the concentration compactness techniques to prove existence and stability results of standing waves for nonlinear fractional Schr\"{o}dinger-Choquard equation \[ i\partial_t\Psi + (-\Delta)^{\alpha}\Psi = a |\Psi|^{s-2}\Psi+\lambda \left( \frac{1}{|x|^{N-\beta}} \star |\Psi|^p \right)|\Psi|^{p-2}\Psi\ \ \ \mathrm{in}\ \mathbb{R}^{N+1}, \] where $N\geq 2$, $\alpha\in (0,1)$, $\beta\in (0, N)$, $s\in (2, 2+\frac{4\alpha}{N})$, $p\in [2, 1+\frac{2\alpha+\beta}{N})$, and the constants $a, \lambda$ are nonnegative satisfying $a+\lambda > 0.$ We then extend the arguments to establish similar results for coupled standing waves of nonlinear fractional Schr\"{o}dinger systems of Choquard type. The same argument works for equations with an arbitrary number of combined nonlinearities and when $|x|^{\beta-N}$ is replaced by a more general convolution potential $\mathcal{K}:\mathbb{R}^N\to [0, \infty)$ under certain assumptions. The same arguments can be applied and the results are identical for the case $\alpha=1$ as well.