Semiclassical states for fractional Schrödinger equations with critical
nonlinearities
Abstract
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.