Multiplicity of solutions for fractional Schrödinger systems in ℝ^N

ABSTRACT

In this paper, we deal with the following nonlocal systems of fractional Schrödinger equations $\left\{\begin{array}{ll}{\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u=Q_u(u,v)+\gamma H_u(u,v)& \text{in }{\mathbb{R}}^N\\ {\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}v+W(x)v=Q_v(u,v)+\gamma H_v(u,v)& \text{in }{\mathbb{R}}^N\\ u,v>0& \text{in }{\mathbb{R}}^N,\end{array}\right.$ where $\epsilon >0$, $s\in (0,1)$, N>2s, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $W:{\mathbb{R}}^N\to \mathbb{R}$ are continuous potentials, Q is a homogeneous $C^2$-function with subcritical growth, $\gamma \in \{0,1\}$ and $H(u,v)=\left(2/\right(\alpha +\beta \left)\right)|u{|}^{\alpha }|v{|}^{\beta }$ with $\alpha ,\beta \ge 1$ such that $\alpha +\beta =2_s^{\ast }$. We investigate the subcritical case $(\gamma =0)$ and the critical case $(\gamma =1)$, and using Ljusternik–Schnirelmann theory, we relate the number of solutions with the topology of the set where the potentials V and W attain their minimum values.

COMMUNICATED BY:

• G. Molica Bisci

KEYWORDS:

• Fractional Schrödinger systems
• variational methods
• Ljusternik–Schnirelmann theory
• positive solutions

AMS SUBJECT CLASSIFICATIONS:

• 35J50
• 35R11
• 58E05

Disclosure statement

No potential conflict of interest was reported by the author.

Acknowledgements

The author would like to express his sincere gratitude to the anonymous referee for her/his valuable comments and suggestions.