Existence of the gauge for fractional Laplacian Schrödinger operators

Abstract

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