On singular equations with critical and supercritical exponents

Abstract

We study the problem(Iε){−Δu−μu|x|2=up−εuqin Ω,u>0in Ω,u∈H01(Ω)∩Lq+1(Ω), 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)=|x|−(p+1)νvp−ε|x|−(q+1)νvqin Ω,v>0in Ω,v∈H01(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.