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 $(-\triangle )^{\alpha /2}$ 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

$$\begin{aligned} (-\triangle )^{\alpha /2} u = u \omega + \nu \,\,\, \text{ on } \,\,\, \Omega , \,\,\, u=0 \,\,\, \text{ on } \,\,\, \Omega ^c, \end{aligned}$$

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

$$\begin{aligned} (-\triangle )^{\alpha /2} u = u \omega \,\,\, \text{ on } \,\,\, \Omega , \,\,\, u \ge 0 \,\,\, \text{ on } \,\,\, \Omega , \,\,\, u=1 \,\,\, \text{ on } \,\,\, \Omega ^c . \end{aligned}$$

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 $1< \alpha <2$. The condition $\Vert T \Vert \le 1$ is necessary for the existence of $u_1$ for all $0<\alpha \le 2$.