Existence of positive solutions for generalized fractional Brézis-Nirenberg problem

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\R^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation} \label{Pr1} (-\Delta_{p})^{s}u - \lambda w |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in } \mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big) , \tag{P} \end{equation} where $s\in (0,1), p\in (1, {N}/{s})$, $p_{s}^{*}={Np}/({N-sp})$ and the operator $(-\Delta_{p})^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big)$ is the completion of $C_c^\infty\big(\R^N\big)$ with respect to the Gagliardo semi-norm. In this article, we prove the existence of a positive solution to problem \eqref{Pr1} by allowing the Hardy weight function $w$ to change its sign.