Estimates of sub and super solutions of Schrödinger equations with very singular potentials

Abstract

Consider operators $L_V:=\Delta +V$ in a bounded smooth domain $D$ in $R^N$. Assume that $V\in C^1(D)$ and $V$ may blow up at the boundary at most as $1/{\delta }^2$ where $\delta$ denotes distance to the boundary. Assume also that $L_V$ has a ground state ${\Phi }_V$ that satsifies an additional condition on its behavior near the boundary (see Section 3). These conditions are satisfied by a large class of potentials (see Section 6). We derive sharp, two-sided estimates of weighted integrals of positive $L_V$ harmonic functions and $L_V$ potentials. These lead to a-priori estimates of positive $L_V$ supersolutions and subsolutions assuming (in the latter case) existence of $L_V$ boundary trace.

This replaces version 1 in which an error occured. In the present version the estimates are derived under a different set of assumptions

References in corpus (2)

• Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains
• Boundary singularities of semilinear elliptic equations with Leray-Hardy potential

Cited by in corpus (1)

• Green kernel and Martin kernel of Schrödinger operators with singular potential and application to the B.V.P. for linear elliptic equations