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

2019/12/03 by Marcus, Moshe
#35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.01283

Abstract

Consider operators L_V:=Δ+ V in a bounded smooth domain D in R^N. Assume that V∈ C¹(D) and V may blow up at the boundary at most as 1/δ² where δ denotes distance to the boundary. Assume also that L_V has a ground state Φ_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.

Related

• Schrodinger equations with very singular potentials in Lipschitz domains
• Schrödinger equations with singular potentials: linear and nonlinear boundary value problems
• Boundary value problems with signed measure data for semilinear Schrödinger equations
• Uniform Sobolev estimates in ℝⁿ involving singular potentials
• Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin