Abstract We consider the following nonlinear Neumann problem { − Δ u − γ u | x | 2 + μ u = | u | 2 s ⁎ − 2 u | x | s in B R ⊂ R N , N ≥ 3 ∂ u ∂ ν = 0 on ∂ B R where γ γ ‾ : = ( N − 2 ) 2 4 , 0 s 2 , 2 s ⁎ = 2 ( N − s ) N − 2 and B R is the ball centered at the origin with radius R. Firstly, we establish the existence of infinitely many positive radial solutions which are singular at the origin. Secondly, we investigate the existence and regularity of a least-energy solution. Lastly, we study the symmetric properties of a regular least-energy solution.

中文翻译：

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具有哈代势的半线性椭圆问题解的性质

摘要 我们考虑以下非线性 Neumann 问题 { − Δ u − γ u | × | 2 + μ u = | 你| 2 s ⁎ − 2 u | × | s in BR ⊂ RN , N ≥ 3 ∂ u ∂ ν = 0 on ∂ BR 其中 γ γ ‾ : = ( N − 2 ) 2 4 , 0 s 2 , 2 s ⁎ = 2 ( N − s ) N − 2 和BR 是圆心在原点，半径为 R 的球。首先，我们建立无穷多个在原点奇异的正径向解的存在性。其次，我们研究了最小能量解的存在性和规律性。最后，我们研究了规则最小能量解的对称性质。