A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents

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In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, 0.1 & Δu + λu²*⁽ˢ¹⁾⁻¹|x|ˢ¹ + u²*⁽ˢ²⁾⁻¹|x|ˢ² =0 in Ω, & u=0 on Ω, where 0 ≤ s₂ < s₁ ≤ 2, 0 ≠ λ∈ 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 0.1 has a least-energy solution. Therefore, this paper has completed the study of 0.1 for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of 0.1 with Ω= under different situations of s₁, s₂ and λ.