A simple construction of a fundamental solution for the extended Weinstein equation

Sirkka-Liisa Eriksson; Heikki Orelma

doi:10.56754/0719-0646.2602.341

DOI:

https://doi.org/10.56754/0719-0646.2602.341

Abstract

In this article, we study the extended Weinstein equation
\[
Lu=\Delta u+\frac{k}{x_n}\frac{\partial u}{\partial x_n}+\frac{\ell}{x_n^2}u,
\]
where \(u\) is a sufficiently smooth function defined in \(\mathbb{R}^n\) with \(x_n>0\) and \(n\ge 3\). We find a detailed construction for a fundamental solution for the operator \(L\). The fundamental solution is represented by the associated Legendre functions \(Q_\nu^\mu\).

Keywords

Extended Weinstein equation , fundamental solution , associated Legendre function

Mathematics Subject Classification:

22E70 , 35A08

Sirkka-Liisa Eriksson University of Helsinki, Helsinki, Finland. https://orcid.org/0000-0002-3592-3716
Heikki Orelma Tampere Institute of Mathematics, Tampere, Finland. https://orcid.org/0000-0002-8251-4333

Pages: 341–358
Date Published: 2024-08-19
Vol. 26 No. 2 (2024)

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