On the generalised Brézis–Nirenberg problem

For p∈ (1 , N) and a domain Ω in RN, we study the following quasi-linear problem involving the critical growth: -Δpu-μg|u|p-2u=|u|p∗-2uinDp(Ω),where Δ p is the p-Laplace operator defined as Δ p(u) = div (| ∇ u| p-2∇ u) , p∗=NpN-p is the critical Sobolev exponent and Dp(Ω) is the Beppo-Levi space defined as the completion of Cc∞(Ω) with respect to the norm ‖u‖Dp:=[∫Ω|∇u|pdx]1p. In this article, we provide various sufficient conditions on g and Ω so that the above problem admits a positive solution for certain range of μ. As a consequence, for N≥ p2, if g is such that g+≠ 0 and the map u↦ ∫ Ω| g| | u| pd x is compact on Dp(Ω) , we show that the problem under consideration has a positive solution for certain range of μ. Further, for Ω = RN, we give a necessary condition for the existence of positive solution.