Semiclassical states of a linearly coupled critical fractional Schrödinger system

Abstract

This paper focuses on the linearly coupled critical fractional Schrödinger system \begin{equation*} \begin{cases} ε^{2s}(-\triangle)^s u +a(x)u=u^p+λv\quad &\text{in}\ \mathbb{R}^N,\\ ε^{2s}(-\triangle)^s v +b(x)v=v^{2_s^*-1}+λu\quad &\text{in}\ \mathbb{R}^N, \end{cases} \end{equation*} where $N>2s,$ $s\in (0,1),$ $p\in (1,2_s^{\ast }),$ $\epsilon$ and $\lambda$ are positive parameters, $a,b\in C({\mathbb{R}}^N)$ are positive potentials, and $(-△{)}^s$ is the fractional Laplacian operator. Under certain assumptions on $a$ and $\lambda ,$ we obtain the existence, decay estimates and concentration property of positive vector ground states for small $\epsilon .$ Furthermore, under an additional assumption on potentials $a$ and $b$, we consider the multiplicity of positive vector solutions for small $\epsilon$, which turn out to have similar decay estimate and concentration property to those of the ground state for small $\epsilon$.

49 pages

References in corpus (1)

• From the long jump random walk to the fractional Laplacian