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

Abstract

We prove the existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$(-\Delta {)}^{s}u=\vartheta \frac{u}{|x{|}^{2s}}+u^{2_s^{\ast }-1},u\in {\dot{H}}^s({\mathbb{R}}^N).$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using of the moving plane method, in a nonlocal setting, on the whole ${\mathbb{R}}^N$ and by 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.

References in corpus (1)

• Symmetry via antisymmetric maximum principles in nonlocal problems of variable order