2022/05/17 by Anoop, T. V., Das, Ujjal
#35B33 #35J60 #58E30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2205.08526
For p ∈ (1,N) and a domain Ω in ℝN, we study the following quasi-linear problem involving the critical growth: -Δp u - μg|u|p-2u = |u|^p*-2u in Dp(Ω), where Δp is the p-Laplace operator defined as Δp(u) = div(|∇ u|p-2 ∇ u), p*= (Np)/(N-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|p dx ]^ (1)/(p). 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|p dx 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.