Hermite elliptic equations
type: Item
instance of: scholarly article
title: Hermite elliptic equations
zbMATH Open document ID: 1019.33007
DOI: 10.1016/S0252-9602(17)30468-X
author: Zhong Ding
published in: Acta Mathematica Scientia. Series B. (English Edition)
publication date: 2003-09-15T00:00:00Z
review text: The following eigen-value problem of the Hermite nonlinear elliptic equation \[ -\Delta u+x^2u=\lambda f(x,u),\quad u\in H^1(\mathbb R^n),\tag{*} \] where \(f(x,u)\) is nondecreasing, Lipschitz and \(|f(x,t)|\leq c|t|^q\), \(1<q<N-1\), \(N=2n/(n-2)\) is considered. The problem is treated by an inverse power method adopted by the author to unbounded domains. It is proved (Theorem 7) that there exists a weak solution \(u\in H^1_{x^2}\) (\(\|u\|^2=\int|\nabla u|^2+x^2u^2 dx\)) of the problem (*). In the linear case \(f(x,u)=f(x),\lambda=1 \) an explicit series solution to the problem (*) is given.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1810183
zbMATH Keywords: Hermite elliptic equation
zbMATH Keywords: inverse power method
zbMATH Keywords: Sobolev type inequalities
zbMATH Keywords: weak solution
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
full work available at URL: s0252-9602(17)30468-x
OpenAlex ID: W2784079335
description: scientific article; zbMATH DE number 1810183