In this paper, we consider the following fractional Schrödinger equation ε 2 s ( − ∆ ) s u + V ( x ) u = P ( x ) f ( u ) + Q ( x ) | u | 2 s ∗ − 2 u in R N , where ε>0 is a parameter, s∈(0 ,1), 2 s ∗ = 2 N N − 2 s , N>2 s, ( − ∆ ) s is the fractional Lapalacian and f is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration-compactness principle, we prove the existence of a positive solution for the above equations.

In this paper, we first study the existence of ground state solutions for the following Schrödinger systems { − ∆ u + V ∞ u = G u ( u , v ) , x ∈ R N , − ∆ v + V ∞ v = G v ( u , v ) , x ∈ R N , u , v > 0 , u , v ∈ H 1 ( R N ) , where N≥3 and G ∈ C 2 ( ( R + ) 2 , R ) . And then, by using variational method and projections on Nehari-Poho z ̆ aev type manifold, we will prove the nonexistence of ground state solutions for the coupled Schrödinger systems { − ∆ u + V ( x ) u = G u ( u , v ) , x ∈ R N , − ∆ v + V ( x ) v = G v ( u , v ) , x ∈ R N , u , v > 0 , u , v ∈ H 1 ( R N ) .