Nondegeneracy of positive solutions to nonlinear Hardy-Sobolev equations

Frédéric Robert (1)

1 IECL - Institut Élie Cartan de Lorraine

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

Domaines: Equations aux dérivées partielles [math.AP]

Frédéric Robert. Nondegeneracy of positive solutions to nonlinear Hardy-Sobolev equations. Advances in Nonlinear Analysis, 2017, A festschrift in honor of Professor Patrizia Pucci’s 65th birthday, 6 (2), pp.237-242. ⟨10.1515/anona-2016-0267⟩. ⟨hal-01412909v2⟩