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Solitary waves for the power degenerate NLS -- existence and stability

Vishnu Iyer, Atanas G. Stefanov

Abstract

We consider a semilinear Schrödinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter $a$ and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness.

Solitary waves for the power degenerate NLS -- existence and stability

Abstract

We consider a semilinear Schrödinger equation, driven by the power degenerate second order differential operator . We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness.

Paper Structure

This paper contains 29 sections, 20 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Standing wave solutions
  3. Linearized problem
  4. Existence results
  5. Stability results
  6. Preliminaries
  7. Function spaces
  8. Caffarelli-Kohn-Nirenberg inequality
  9. Necessary and sufficient conditions for stability of the eigenvalue problem \ref{['l:20']}
  10. Variational construction
  11. The compactness of the embedding $H^{1,a}({\mathbb R}^d)\hookrightarrow L^q({\mathbb R}^d)$
  12. Existence of the waves
  13. The existence claim in Theorem \ref{['theo:10']}
  14. Stability of the waves
  15. The linearized operators ${\mathcal{L}}_\pm$ and their spectral properties
  16. ...and 14 more sections

Key Result

Theorem 1

Let $d\geq 1$ and $a, p>1$ satisfy the relation Then, the equation 120 has a distributional positive solution $\phi: \phi\in C^\infty({\mathbb R}^d\setminus\{0\})$. Moreover, $\phi\in H^{1,a}({\mathbb R}^d)$ (see the definition h1a below), and there is the following pointwise exponential bound for some $\delta>0, C>0$. In case the solution $\phi$ is radial, then it is also continuous at zero,

Theorems & Definitions (31)

  • Theorem 1
  • Proposition 1
  • Corollary 1
  • Theorem 2
  • Definition 1.1
  • Theorem 3
  • Theorem 4
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 21 more