标题： 径向对称性，退化非线性薛定谔方程解的唯一性和非退化性 显示英文标题

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

摘要： 在本文中，我们考虑退化的非线性椭圆方程$$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + \omega u=|u|^{p-2}u \quad \mbox{in} \,\, \mathbb{R}^d, $$的解的径向对称性、唯一性和非退化性，其中$d \geq 2$，$0<a<1$，$\omega>0$和$2<p<\frac{2d}{d-2(1-a)}$。我们证明了任何基态都是径向对称的，并且在径向方向上严格递减。此外，我们建立了基态的唯一性，并推导了相应径向对称 Sobolev 空间中基态的非退化性。这证实了最近在\cite{IS}中提出的自然猜想。 显示英文摘要

摘要： 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}.