Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem

Abstract

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain ${\mathbb{R}}^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation*} (-Δ)_{p}^{s}u - λw |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in} ~\mathcal{D}^{s,p}(\mathbb{R}^{N}), \end{equation*} where $s\in (0,1),p\in (1,\frac{N}{s}),p_s^{\ast }=\frac{Np}{N-sp}$ and the operator $(-\Delta {)}_p^s$ is the fractional $p$-Laplace operator. The space ${\mathcal{D}}^{s,p}({\mathbb{R}}^N)$ is the completion of $C_c^{\infty }({\mathbb{R}}^N)$ with respect to the Gaglairdo semi-norm. In this article, we prove the existence of a positive solution to this problem by allowing the Hardy weight $w$ to change its sign.

28 pages