Sharp existence results on fractional elliptic equation

We consider the following mass-constrained elliptic problem $\left(\begin{array}{c}{(-\Delta )}^{s}u+V\left(x\right)u+\lambda u=f(x,u)in{R}^N,\\ {\int }_{R^N}{\left|u\left(x\right)\right|}^{2}dx=c,\end{array}\right)$with $0<s<1$ and ${(-\Delta )}^s$ is fractional Laplacian. We get a sharp description of the existence and non-existence of the global minimizer on the mass constraint, which is called energy ground state. More specifically, we show that there exists a constant $c_0$ such that there exists an energy ground state if $c>c_0$ and there exists no energy ground state if $0<c<c_0$. Our results extends some related works.