Singularly Perturbed Fractional Schrödinger Equations with Critical Growth

We are concerned with the following singularly perturbed fractional Schrödinger equation:\left\{\begin{aligned} &\displaystyle{\varepsilon^{2s}}{(-\Delta)^{s}}u+V(x)u=% f(u)&&\displaystyle{\text{in }}{\mathbb{R}^{N}},\\ &\displaystyle u\in{H^{s}}({\mathbb{R}^{N}}),&&\displaystyle u>0{\text{ on }}{% \mathbb{R}^{N}},\end{aligned}\right.where ε is a small positive parameter,{N>2s}, and{{(-\Delta)^{s}}}, with{s\in(0,1)}, is the fractional Laplacian. Using variational technique, we construct a family of positive solutions{{u_{\varepsilon}}\in{H^{s}}({\mathbb{R}^{N}})}which concentrates around the local minima ofVas{\varepsilon\to 0}under general conditions onfwhich we believe to be almost optimal.