On the Dirichlet problem for the Schrödinger equation in the upper half-space

被引：0
作者：
Bo Li
Tianjun Shen
Jian Tan
Aiting Wang
机构：
[1] Jiaxing University,College of Data Science
[2] Tianjin University,Center for Applied Mathematics
[3] Nanjing University of Posts and Telecommunications,School of Science
[4] Qinghai Minzu University,School of Mathematics and Statistics
来源：
Analysis and Mathematical Physics | 2023年 / 13卷
关键词：
Boundary value problem; Elliptic equation; Morrey function; Regularity; Variable exponent; Primary 35J25; Secondary 42B35;
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摘要：
A well-known result of Stein-Weiss in 1971 said that a harmonic function, defined on the upper half-space, is the Poisson integral of a Lebesgue function if and only if it is also a Lebesgue function uniformly in the time variable. Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed.
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