Existence of the gauge for fractional Laplacian Schrödinger operators

Abstract

Let $\Omega \subseteq {\mathbb{R}}^n$ be an open set, where $n\ge 2$. Suppose $\omega$ is a locally finite Borel measure on $\Omega$. For $\alpha \in (0,2)$, define the fractional Laplacian $(-△{)}^{\alpha /2}$ 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 $\Vert T{\Vert }_{L^2(\omega )\to L^2(\omega )}<1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L_0^{\alpha /2,2}(\Omega )$ of \[ (-\triangle)^{α/2} u = u ω+ ν\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, Ω^c, \] 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 \[ (-\triangle)^{α/2} u = u ω\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, Ω^c . \] We show that if $\Vert T\Vert <1$, then $u_1$ always exists if $0<\alpha <1$. For $1\le \alpha <2$, 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 $1<\alpha <2$.

31 pages