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arXiv · 2503.00708

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

Tianxiang Gou
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Abstract

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + ωu=|u|^{p-2}u \quad \mbox{in} \,\, \R^d, $$ where $d \geq 2$, $0 0$ 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}.

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Tianxiang Gou. 2026-04-14. Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schrödinger equations. https://arxiv.org/abs/2503.00708

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