2011/02/21 by YanYan Li, Chang-Shou Lin · 2 citations
Mathematics · #math.AP #msc:35J60
paper · pdf · doi:10.1007/s00205-011-0467-2
arxiv created 2011/02/21 · arxiv updated 2015/05/27
In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, & Δu + λ\fracu2^*(s1)-1|x|s1 + \fracu2^*(s2)-1|x|s2 =0 in Ω, & u=0 on Ω, where 0 ≤ s2 < s1 ≤ 2, 0 ≠ λ∈ \Bbb R and 0 ∈ ∂ Ω. The existence (or nonexistence) for least-energy solutions has been extensively studied when s1=0 or s2=0. In this paper, we prove that if 0< s2 < s1 <2 and the mean curvature of ∂ Ω at 0 H(0)<0, then \eqref0.1 has a least-energy solution. Therefore, this paper has completed the study of \eqref0.1 for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of \eqref0.1 with Ω=\rn under different situations of s1, s2 and λ.