Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schr\'odinger equations

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + \omega u=|u|^{p-2}u \quad \mbox{in} \,\, \mathbb{R}^d, $$ where $d \geq 2$, $0<a<1$, $\omega>0$ and $2<p<\frac{2d}{d-2(1-a)}$. We proved that any ground state is radially symmetric and strictly decreasing in the radial direction. Moreover, we establish the uniqueness of ground states and derive the non-degeneracy of ground states in the corresponding radially symmetric Sobolev space. This affirms the nature conjectures posed recently in \cite{IS}.
Comment: 18 pages