Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schrödinger equations

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$-\mathrm{\nabla }\cdot \left(\mathrm{\mid }x{\mathrm{\mid }}^{2a}\mathrm{\nabla }u\right)+\omega u=\mathrm{\mid }u{\mathrm{\mid }}^{p-2}uin{\mathbb{R}}^d,$$ where $d\ge 2$, $0<a<1$, $\omega >0$ 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 natural conjectures posed recently in \cite{IS}.