Positive solutions and harmonic measure for Schrödinger operators in uniform domains
We give bilateral pointwise estimates for positive solutions of the equation \begin{equation*} \left\{ \begin{aligned} -\triangle u & = ωu \, \,& & \mbox{in} \, \, Ω, \quad u \ge 0, \\ u & = f \, \, & &\mbox{on} \, \, \partial Ω, \end{aligned} \right. \end{equation*} in a bounded uniform domain $Ω\subset {\bf R}^n$, where $ω$ is a locally finite Borel measure in $Ω$, and $f\ge 0$ is integrable with respect to harmonic measure $d H^{x}$ on $\partialΩ$. We also give sufficient and matching necessary conditions for the existence of a positive solution in terms of the exponential integrability of $M^{*} (m ω)(z)=\int_ΩM(x, z) m(x)\, d ω(x)$ on $\partialΩ$ with respect to $f \, d H^{x_0}$, where $M(x, \cdot)$ is Martin's function with pole at $x_0\in Ω, m(x)=\min (1, G(x, x_0))$, and $G$ is Green's function. These results give bilateral bounds for the harmonic measure associated with the Schrödinger operator $-\triangle - ω$ on $Ω$, and in the case $f=1$, a criterion for the existence of the gauge function. Applications to elliptic equations of Riccati type with quadratic growth in the gradient are given.