Existence of positive solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

Abstract

In this paper, we have used variational methods to study existence of solutions for the following critical nonlocal fractional Hardy elliptic equation \\begin{equation*} (- \\Delta)^s u - \\gamma \\frac{u}{|x|^{2 s}} = \\frac{|u|^{2_s^*(b) - 2} u}{|x|^{b}} + \\lambda f (x, u ),\\quad \\text{in } \\mathbb{R}^N, \\end{equation*} where $N > 2 s $, $ 0< s< 1 $, $ \\gamma, \\lambda $ are real parameters, $(- \\Delta)^s$ is the fractional Laplace operator, $2_s^*(b) = {2 (N - b)}/(N - 2s)$ is a critical Hardy-Sobolev exponent with $b \\in [0, 2s)$ and $ f \\in C(\\mathbb{R^{N}} \\times \\mathbb{R}, \\mathbb{R})$.