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

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Abstract

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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Notes

  1. In fact, this assumption is more or less different from the corresponding assumption of [21] in form, but they are coincident with each other.

  2. When the potential v is non-trivial, the sharp upper bound of \(h_t^v(x,y)\) may be controlled by the classical heat kernel with an additional exponential decay term; see [42] for example.

  3. When the potential v is non-trivial, there are two points should to be mentioned. On the one hand, the boundedness for an \(\mathbb {L}\)-harmonic function may be relaxed to the polynomial growth; see Examples 2.13 and 2.14 for more details. On the other hand, the “constant” in the conclusion must be zero due to the structure of the elliptic equation.

  4. Note that we can improve the critical reverse Hölder index \((n\text {+}1)/2\) to n/2 through some totally different techniques.

  5. In [37], the author assume \(n\ge 3\).

  6. If \(n\,\hbox {=}\,1\) or \(n\,\hbox {=}\,2\), the Kato condition (K) is superfluous. We can alone assume that the potential v is in the Muckenhoupt class \(A_\infty \).

  7. We thanks Prof. Kangwei Li provide the counter-example \(v(x)\,\hbox {=}\,|x|^{-2}\).

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Funding

Bo Li is supported by National Natural Science Foundation of China (12201250) and Natural Science Foundation of Zhejiang Province of China (LQ23A010007). Jian Tan is supported by National Natural Science Foundation of China (11901309), China Postdoctoral Science Foundation (2023T160296), and Natural Science Foundation of Nanjing University of Posts and Telecommunications (NY222168).

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Authors and Affiliations

  1. College of Data Science, Jiaxing University, Jiaxing, 314001, China

    Bo Li

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Tianjun Shen

  3. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Jian Tan

  4. School of Mathematics and Statistics, Qinghai Minzu University, Xining, 810000, China

    Aiting Wang

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  1. Bo Li
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  2. Tianjun Shen
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  4. Aiting Wang
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All authors contributed equality and significantly in writing this paper. All authors read and approved the final manuscript.

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Correspondence to Jian Tan.

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Li, B., Shen, T., Tan, J. et al. On the Dirichlet problem for the Schrödinger equation in the upper half-space. Anal.Math.Phys. 13, 85 (2023). https://doi.org/10.1007/s13324-023-00834-6

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