Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential

Abstract

We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$\begin{aligned} (-\Delta )^s u=\vartheta \frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb {R}^N), \quad N> 2s,\quad 0<s<1. \end{aligned}$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using the moving plane method, in a nonlocal setting, on the whole \(\mathbb {R}^{N}\) and some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem into a different nonlocal problem in a weighted fractional space.