Multiplicity of solutions for fractional Schrödinger systems in ${\mathbb{R}}^N$

Abstract

In this paper we deal with the following nonlocal systems of fractional Schrödinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-Δ)^{s}u+V(x)u=Q_{u}(u, v)+γH_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\\ \varepsilon^{2s} (-Δ)^{s}v+W(x)v=Q_{v}(u, v)+γH_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} \\ u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\epsilon >0$, $s\in (0,1)$, $N>2s$, $(-\Delta {)}^s$ 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 $H(u,v)=\frac{2}{\alpha +\beta }|u{|}^{\alpha }|v{|}^{\beta }$ 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.

References in corpus (1)

• Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation