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

In this article, we study the fractional Br\'{e}zis-Nirenberg type problem on whole domain RN associated with the fractional p-Laplace operator. To be precise, we want to study the following problem: {equation*} (-Δ)ₚˢu - λ w |u|ᵖ⁻²u= |u|^{pₛ*-2}u in ~Ds,p(RN), {equation*} where s∈ (0,1),~p ∈ (1,N/s), ~pₛ*= Np/N-sp and the operator (-Δ)ₚˢ is the fractional p-Laplace operator. The space Ds,p(RN) is the completion of Cc^∞(RN) 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.