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Existence of the gauge for fractional Laplacian Schrödinger operators. (English) Zbl 1472.35434

Summary: 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 \((-\Delta)^{\alpha/2}\) 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 \[ (-\Delta)^{\alpha/2} u=u\omega+\nu\text{ on }\Omega,\quad u=0\text{ on }\Omega^c, \] 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 \[ (-\Delta)^{\alpha/2}u=u\omega\text{ on }\Omega,\quad u\geq 0\text{ on }\Omega,\quad u=1\text{ on }\Omega^c. \] 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 \(1<\alpha<2\). The condition \(\Vert T\Vert\leq 1\) is necessary for the existence of \(u_1\) for all \(0<\alpha\leq 2\).

MSC:

35R11 Fractional partial differential equations
31B35 Connections of harmonic functions with differential equations in higher dimensions
35B45 A priori estimates in context of PDEs
35J08 Green’s functions for elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
35J25 Boundary value problems for second-order elliptic equations
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