Existence of Positive Solutions for Generalized Fractional Br\'{e}zis-Nirenberg Problem

Abstract

In this article, we study the fractional Br\'{e}zis-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*} (-\Delta)_{p}^{s}u - \lambda w |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in} ~\mathcal{D}^{s,p}(\mathbb{R}^{N}), \end{equation*} where $s\in (0,1),~p \in (1,\frac{N}{s}), ~p_{s}^{*}= \frac{Np}{N-sp}$ 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.