Existence of the gauge for fractional Laplacian Schr\"{o}dinger operators

Abstract

Let $\Omega \subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. Suppose $\omega $ is a locally finite Borel measure on $\Omega$. For $\alpha \in (0,2)$, define the fractional Laplacian $(-\triangle )^{\alpha/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $\alpha$ on $\Omega$. Define $T(u) = G(u \omega).$ If $\Vert T \Vert_{L^2(\omega) \rightarrow L^2 (\omega)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{\alpha/2, 2}_0 (\Omega)$ of \[ (-\triangle)^{\alpha/2} u = u \omega + \nu \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, \Omega^c, \] for $\nu$ in the dual Sobolev space $L^{-\alpha/2, 2} (\Omega)$. If $\Omega$ is a bounded $C^{1,1}$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $\nu = \chi_{\Omega}$. These estimates are used to study the existence of a solution $u_1$ (called the "gauge") of the integral equation $u_1=1+G(u_1 \omega)$ corresponding to the problem \[ (-\triangle)^{\alpha/2} u = u \omega \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, \Omega^c . \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<\alpha <1$. For $1 \leq \alpha <2$, a solution exists if the norm of $T$ is sufficiently small. We also show that the condition $\Vert T \Vert <1$ does not imply the existence of a solution if $1 < \alpha <2$.