Profile of solutions for nonlocal equations with critical and supercritical nonlinearities

Abstract

We study the fractional Laplacian problem [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is a parameter. Here, [Formula: see text] is a bounded star-shaped domain with smooth boundary and [Formula: see text]. We establish existence of a variational positive solution [Formula: see text] and characterize the asymptotic behavior of [Formula: see text] as [Formula: see text]. When [Formula: see text], we describe how the solution [Formula: see text] blows up at an interior point of [Formula: see text]. Furthermore, we prove the local uniqueness of solution of the above problem when [Formula: see text] is a convex symmetric domain of [Formula: see text] with [Formula: see text] and [Formula: see text].