Positive solutions to Schr\"odinger's equation and the exponential integrability of the balayage

TL;DR

This paper establishes necessary and sufficient boundary conditions involving exponential integrability of the balayage measure for the existence of positive solutions to Schrödinger's equation with potential q, providing sharp criteria and boundary estimates.

Contribution

It introduces a new boundary condition based on exponential integrability of the balayage of the measure δq dx, solving an open problem and extending the understanding of Schrödinger equations with nonnegative potentials.

Findings

Necessary and sufficient conditions for solution existence.

Sharp boundary estimates for solutions.

Connection between balayage exponential integrability and solution criteria.

Abstract

Let $\Omega \subset \mathbb{R}^n$, for $n \geq 2$, be a bounded $C^2$ domain. Let $q \in L^1_{loc} (\Omega)$ with $q \geq 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for the existence of very weak solutions to the boundary value problem $(-\triangle -q) u =0, \, \, u\ge 0 \, \, \text{on} \, \, \Omega, \, u=1 \, \text{on} \, \, \partial \Omega$, and the related nonlinear problem with quadratic growth in the gradient, $-\triangle u = |\nabla u|^2 + q \, \text{on} \, \Omega, \, u=0 \, \, \text{on} \, \, \partial \Omega$. We also obtain precise pointwise estimates of solutions up to the boundary. A crucial role is played by a new "boundary condition" on $q$ which is expressed in terms of the exponential integrability on $\partial \Omega$ of the balayage of the measure $\delta q \, dx$, where $\delta (x) = \text{dist} (x, \partial \Omega)$. This condition is sharp, and appears in such a context for the first time. It holds, for example, if $\delta q \, dx$ is a Carleson measure in $\Omega$, or if its balayage is in $BMO(\partial \Omega)$, with sufficiently small norm. This solves an open problem posed in the literature.