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On solutions to $-Δu = V u$ near infinity

Yifei Pan, Yuan Zhang

TL;DR

The paper investigates unique continuation and sign-changing behavior for solutions of $-\,\Delta u = V u$ near infinity under controlled growth of the potential $V$. It develops a Kelvin-transform approach to convert infinity problems into local questions near the origin, and uses a Li–Nirenberg-type monotonicity framework along with Harvey–Polking extension to establish that solutions vanishing to infinite order at infinity must vanish identically; it also identifies conditions under which nontrivial bounded solutions must change sign near infinity, with implications for homogeneous radial potentials and nonzero eigenvalues. The results extend the landscape around the Landis conjecture by providing explicit growth/monotonicity assumptions that yield sharp unique continuation and sign-change phenomena, complemented by counterexamples that demonstrate the necessity of the hypotheses.

Abstract

We investigate the unique continuation property and the sign changing behavior of weak solutions to $-Δu =Vu$ near infinity under certain conditions on the blow-up rate of the potential $V$ near infinity.

On solutions to $-Δu = V u$ near infinity

TL;DR

The paper investigates unique continuation and sign-changing behavior for solutions of near infinity under controlled growth of the potential . It develops a Kelvin-transform approach to convert infinity problems into local questions near the origin, and uses a Li–Nirenberg-type monotonicity framework along with Harvey–Polking extension to establish that solutions vanishing to infinite order at infinity must vanish identically; it also identifies conditions under which nontrivial bounded solutions must change sign near infinity, with implications for homogeneous radial potentials and nonzero eigenvalues. The results extend the landscape around the Landis conjecture by providing explicit growth/monotonicity assumptions that yield sharp unique continuation and sign-change phenomena, complemented by counterexamples that demonstrate the necessity of the hypotheses.

Abstract

We investigate the unique continuation property and the sign changing behavior of weak solutions to near infinity under certain conditions on the blow-up rate of the potential near infinity.
Paper Structure (4 sections, 21 theorems, 107 equations)

This paper contains 4 sections, 21 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unique continuation property at infinity
  4. Sign changing property near infinity

Key Result

Theorem 1.1

Let $V\in L_{loc}^{\infty}(\mathbb R^n)$. Assume further that when $|x|>>1$, $V$ is locally Lipschitz, for some constant $M\in \mathbb R$, and in terms of the polar coordinates $(r, \theta)$. Let $u\in L^2_{loc}(\mathbb R^n)$ be a weak solution to If $\lim_{r\rightarrow \infty}r^m\int_{|x|>r} |u|^2 = 0$ for each $m\ge 0$, then $u\equiv 0$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Example 1
  • Lemma 2.1
  • ...and 36 more