Nondegeneracy of positive solutions to nonlinear Hardy–Sobolev equations

Abstract

In this note, we prove that the kernel of the linearized equation around a positive energy solution in ℝn${\mathbb{R}^{n}}$, n≥3${n\geq 3}$, to the problem -Δ⁢W-γ⁢|x|-2⁢V=|x|-s⁢W2⋆⁢(s)-1$-\Delta W-\gamma|x|^{-2}V=|x|^{-s}W^{2^{\star}(s)-1}$ is one-dimensional when s+γ>0$s+\gamma>0$. Here, s∈[0,2)${s\in[0,2)}$, 0≤γ<(n-2)2/4${0\leq\gamma<(n-2)^{2}/4}$ and 2⋆⁢(s)=2⁢(n-s)/(n-2)${2^{\star}(s)=2(n-s)/(n-2)}$.

Keywords

• nondegeneracy
• conformal invariance
• hardy–sobolev equations
• 35j20
• 35j60
• 35j75