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 Schrödinger-Choquard equation \[ i\partial_tΨ+ (-Δ)^αΨ= a |Ψ|^{s-2}Ψ+λ\left( \frac{1}{|x|^{N-β}} \star |Ψ|^p \right)|Ψ|^{p-2}Ψ \ \ \mathrm{in}\ \mathbb{R}^{N+1}, \] where $N\ge 2$, $\alpha \in (0,1)$, $\beta \in (0,N)$, $s\in (2,2+\frac{4\alpha }{N})$, $p\in [2,1+\frac{2\alpha +\beta }{N})$, 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 Schrödinger 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.

31 pages, revised version

Cited by in corpus (9)

• Energy-dependent noncommutative quantum mechanics
• The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials
• Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent
• On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation
• Limit profiles for singularly perturbed Choquard equations with local repulsion
• Existence and stability of standing waves for coupled nonlinear Hartree type equations
• Normalized ground states for the fractional nonlinear Schrödinger equations
• Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
• Semiclassical states of a linearly coupled critical fractional Schrödinger system