On fractional Schrodinger systems of Choquard type

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.