Concentration Phenomena for Fractional Elliptic Equations Involving Exponential Critical Growth

Abstract

Abstract In this paper, we deal with the singular perturbed fractional elliptic problem ε ⁢ ( - Δ ) 1 / 2 ⁢ u + V ⁢ ( z ) ⁢ u = f ⁢ ( u ) $\varepsilon(-\Delta)^{1/2}{u}+V(z)u=f(u)$ in ℝ $\mathbb{R}$ , where ( - Δ ) 1 / 2 ⁢ u ${(-\Delta)^{1/2}u}$ is the square root of the Laplacian and f ⁢ ( s ) ${f(s)}$ has exponential critical growth. Under suitable conditions on f ⁢ ( s ) ${f(s)}$ , we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of V as ε goes to 0.