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 $0.$
Categories: math.AP