Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains. II
type: Item
instance of: scholarly article
title: Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains. II
zbMATH Open document ID: 0682.35044
author: Takashi Suzuki
author: Ken'ichi Nagasaki
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1195513390
review text: [For part I see ibid., Ser. A 64, No. 1, 1--4 (1988; Zbl 0664.35067).] The main object is to show the existence of nonradial solutions of the nonlinear eigenvalue problem (P): \[ -\Delta u=\lambda e^ u\text{ in } \Omega \text{ and } u=0\text{ on } \partial \Omega, \] where \(\Omega\) is an annulus in \({\mathbb R}^ 2.\) After introducing \( K_{\infty}(\mu)=\left\{v\in H^ 1_ 0(\Omega)\mid v\text{ is radial and }\int_{\Omega}e^ v \,dx=\mu\right\}\) and \[ K_ k(\mu)=\left\{v\in H^ 1_ 0(\Omega)\mid v\text{ is invariant under the rotation of }2\pi/k\text{ in } {\mathbb R}^ 2\text{ and } \int_{\Omega}e^ v \,dx=\mu \right\}, \] variational problems \[ j_ k(\mu)=\inf \left \{\int_{\Omega}| \nabla v|^ 2 \,dx\mid v\in K_ k(\mu)\right\}\quad (k=1,2,3,...,\infty) \] are considered. The minimizers \(u_ k\in K_ k(\mu)\) exist and are proved to be solutions of (P). From the classification of critical values \(j_{\infty}(\mu)\) and \(j_ k(\mu)\) \((k=1,2,3,...)\) follows the fact that \(u_ k\) are nonradial for \(\mu\) large enough.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 4117045
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: radial and nonradial solutions
MaRDI profile type: Publication
OpenAlex ID: W1979288056
cites work: Some problems in the theory of quasilinear equations
cites work: Symmetry and related properties via the maximum principle
cites work: Rearrangements and convexity of level sets in PDE
cites work: Q3828418
cites work: Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains. I
cites work: Q3326480
DOI: 10.3792/PJAA.64.29
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description: scientific article; zbMATH DE number 4117045