Symmetric solutions for some elliptic equation on spherical cap of \(\mathrm{S}^n\)
type: Item
instance of: scholarly article
title: Symmetric solutions for some elliptic equation on spherical cap of \(\mathrm{S}^n\)
author: Munemitsu Hirose
published in: Tokyo Journal of Mathematics
publication date: 2024-09-27T00:00:00Z
review text: This paper studied the existence of positive and symmetric solutions for the problem\N\[\N\begin{cases} \Delta_{{\mathbb S}^n}w+\lambda w+w^p=0 \quad \text{in}\ D_\beta\\\Nw>0 \quad \text{in}\ D_\beta\\\Nw=0\quad \text{on}\ \partial D_\beta, \end{cases}\N\]\Nwhere \(\lambda\in R\) and \(P>1\) are parameters, \(D_\beta\) is the spherical cap including the North pole \((0,\ldots, 0, 1)\) on \({\mathcal S}^n\) whose geodesic radius is denoted by \(\beta\in (0,\pi)\), and \(\Delta_{{\mathbb S}^n}\) is the Laplace-Beltrami operator on \({\mathcal S}^n\). The paper investigated non-critical case for \(n\)-dimensional space, and for the sub-critical case, it proves the existence and nonexistence result if the spherical cap is contained in the hemisphere.
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 35B06
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 7919761
zbMATH Keywords: symmetric solution
zbMATH Keywords: boundary value problem
zbMATH Keywords: spherical cap
zbMATH Keywords: Laplace-Beltrami operator
MaRDI profile type: Publication
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DOI: 10.3836/TJM/1502179408
description: scientific article; zbMATH DE number 7919761