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

Tianxiang Gou

Abstract

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + ωu=|u|^{p-2}u \quad \mbox{in} \,\, \mathbb{R}^d, $$ where $d \geq 2$, $0<a<1$, $ω>0$ 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}.

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

Abstract

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation where , , and . 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}.

Paper Structure

This paper contains 4 sections, 14 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Radial Symmetry
  3. Uniqueness
  4. Non-degeneracy

Key Result

Theorem 1.1

(IS) Let $d \geq 2$, $0<a<1$, $\omega>0$ and $2<p<2^*_a$. Then there exists a positive ground state $u \in H^{1,a}(\mathbb{R}^d) \cap C^{\infty}(\mathbb{R}^d \backslash \{0\})$ to equ satisfying the pointwise exponential bound for some $\delta>0$. Moreover, if the solution $u$ is radially symmetric, then it is continuous at zero and satisfies that where $u^{p-1}(0)-\omega u(0)>0$. In particular,

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • ...and 17 more