Concentration phenomena for fractional elliptic equations involving exponential critical growth

Abstract

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