Positive solutions to Schrödinger's equation and the exponential integrability of the balayage

Abstract

Let $\Omega \subset {\mathbb{R}}^n$, for $n\ge 2$, be a bounded $C^2$ domain. Let $q\in L_{loc}^1(\Omega )$ with $q\ge 0$. 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 $(-△-q)u=0,u\ge 0\text{on}\Omega ,u=1\text{on}\partial \Omega$, and the related nonlinear problem with quadratic growth in the gradient, $-△u=|\nabla u{|}^2+q\text{on}\Omega ,u=0\text{on}\partial \Omega$. 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 $\delta qdx$, 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 $\delta qdx$ 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.

33 pages

References in corpus (3)

• Global estimates for kernels of Neumann series and Green's functions
• Pointwise estimates of solutions to nonlinear equations for nonlocal operators
• Pointwise estimates of solutions to semilinear elliptic equations and inequalities

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