Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional Laplacian \begin{gather*}(-\Delta)^{\alpha}u+u=f(x,u)\text{~in~}\mathbb{R}^{N},\\u>0\text{~in~}\mathbb{R}^{N},\qquad\lim_{|x|\rightarrow\infty}u(x)=0.\end{gather*} Furthermore, we analyse the regularity, decay and symmetry properties of these solutions.