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A sharper energy method for the localization of the support to some stationary Schr{ö}dinger equations with a singular nonlinearity

Pascal Bégout, Jesús Ildefonso Díaz

Abstract

We prove the compactness of the support of the solution of some stationary Schr{ö}dinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature and, in particular, by the authors.

A sharper energy method for the localization of the support to some stationary Schr{ö}dinger equations with a singular nonlinearity

Abstract

We prove the compactness of the support of the solution of some stationary Schr{ö}dinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature and, in particular, by the authors.

Paper Structure

This paper contains 5 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. From suitable local inequalities to the vanishing of the involved complex functions on some small ball
  3. A general framework of applications related to the Schrödinger operator
  4. Proofs of the main results
  5. Application to the localization property to the case of Neumann boundary conditions

Key Result

Theorem 2.1

Assume $0<m<1$ and let $N\in\mathbb{N}.$ Then there exists $C=C(N,m)$ satisfying the following property$:$ let $x_0\in\mathbb{R}^N,$$\rho_0>0$ and $u\in H^1_\mathrm{loc}(B(x_0,\rho_0)).$ If there exist $L>0$ and $M>0$ such that for almost every $\rho\in(0,\rho_0),$ then $u_{|B(x_0,\rho_\mathrm{max})}\equiv0,$ where and where, for any $\tau\in\left(\frac{m+1}{2},1\right].$

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Example 3.3
  • Remark 5.1
  • Corollary 5.3: Neumann boundary conditions
  • Remark 5.5
  • ...and 3 more