On singular equations with critical and supercritical exponents

Abstract

We study the problem (𝐼𝜀)◂{:▸ where 𝑞>𝑝≥2⁎−1, 𝜀>0, Ω⊆ℝ𝑁 is a bounded domain with smooth boundary, 0∈Ω, 𝑁≥3 and 0<𝜇<𝜇¯:=(𝑁−22)2. We completely classify the singularity of solution at 0 in the supercritical case. Using the transformation 𝑣=|𝑥|𝜈𝑢, we reduce the problem (𝐼𝜀) to (𝐽𝜀) (𝐽𝜀)⎧⎩⎨⎪⎪−𝑑𝑖𝑣(|𝑥|−2𝜈∇𝑣)𝑣𝑣=|𝑥|−(𝑝+1)𝜈𝑣𝑝−𝜀|𝑥|−(𝑞+1)𝜈𝑣𝑞in Ω,>0in Ω,∈𝐻10(Ω,|𝑥|−2𝜈)∩𝐿𝑞+1(Ω,|𝑥|−(𝑞+1)𝜈), and then formulating a variational problem for (𝐽𝜀), we establish the existence of a variational solution 𝑣𝜀 and characterize the asymptotic behavior of 𝑣𝜀 as 𝜀→0 by variational arguments when 𝑝=2⁎−1.