Fractional Schrödinger equation

Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well

In this paper, we are concerned with the following eigenvalue problem \begin{equation*} (-\Delta)^su+\lambda g(x)u=\alpha u,\ \ u\in H^s(\mathbb{R}^N),\ N\geq3, \end{equation*} where $s\in(0,1),,\alpha,\lambda\in\mathbb{R}$ and \begin{equation*} g(x)\equiv0\ \text{on}\ \bar{\Omega},\ \ g(x)\in(0,1]\ \text{on}\ \mathbb{R}^N\backslash\bar{\Omega}\ \text{and}\ \lim_{|x|\rightarrow\infty}g(x)=1 \end{equation*} for some bounded open set $\Omega\subset\mathbb{R}^N$.

Three solutions for a fractional Schrödinger equation with vanishing potentials

In this paper, we study the following fractional Schrödinger equation \begin{equation*} (-\Delta)^su+V(x)u=K(x)f(u)+\lambda W(x)|u|^{p-2}u,\ \ x\in\mathbb{R}^N, \end{equation*} where $\lambda>0$ is a parameter, $(-\Delta)^s$ denotes the fractional Laplacian of order $s\in(0,1)$, $N>2s$, $W\in L^{\frac{2}{2-p}}(\mathbb{R}^N,\mathbb{R}^+)$, $1< p <2$, $V,K$ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth.