H\"{o}lder regularity and Liouville Theorem for the Schr\"{o}dinger equation with certain critical potentials, and applications to Dirichlet problems

TL;DR

This paper proves Hölder regularity and Liouville theorems for Schrödinger equations with critical potentials on metric measure spaces, extending regularity results to the critical potential index and applying them to boundary value problems.

Contribution

It establishes Hölder continuity and Liouville theorems for Schrödinger equations with critical potentials on metric measure spaces, identifying the precise critical potential index for regularity.

Findings

Hölder regularity for solutions with potentials in reverse Hölder classes

Liouville theorem for solutions under critical potential conditions

Characterizations of boundary value solutions in BMO/CMO/Morrey spaces

Abstract

Let $(X,d,\mu)$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\ge n>1$, and admitting an $L^2$-Poincar\'e inequality. In this article, we establish the H\"{o}lder continuity and a Liouville-type theorem for the (elliptic-type) Schr\"odinger equation $$\mathbb L u(x,t)=-\partial^2_{t}u(x,t)+\mathcal L u(x,t)+V(x)u(x,t)=0,\quad x\in X,\, t\in\mathbb R, $$ where $\mathcal L$ is a non-negative operator generated by a Dirichlet form on $X$, and the non-negative potential $V$ is a Muckenhoupt weight belonging to the reverse H\"older class ${RH}_q(X)$ for some $q>\max\{Q/2,1\}$. Note that $Q/2$ is critical for the regularity theory of $-\Delta+V$ on $\mathbb{R}^Q$ ($Q\ge3$) by Shen's work in 1995, which hints the critical index of $V$ for the regularity results above on $X\times \mathbb R$ may be $(Q+1)/2$. Our results show that this critical index is in fact $\max\{Q/2,1\}$. Our approach primarily relies on the controllable growth of $V$ and the elliptic theory for the operator $\mathbb L$/$-\partial^2_{t}+\mathcal{L}$ on $X\times \mathbb R$, rather than the analogs for $\mathcal L+V$/$\mathcal{L}$ on $X$, under the critical index setting. As applications, we further obtain some characterizations for solutions to the Schr\"odinger equation $-\partial^2_{t}u+\mathcal L u+Vu=0$ in $X\times \mathbb R_+$ with boundary values in BMO/CMO/Morrey spaces related to $V$, improving previous results to the critical index $q>\max\{Q/2,1\}$.