Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in \({\mathbb{R}}^ 2\)
type: Item
instance of: scholarly article
title: Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in \({\mathbb{R}}^ 2\)
zbMATH Open document ID: 0681.35032
DOI: 10.1007/BF02874647
author: Adimurthi
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1989-01-01T00:00:00Z
review text: Consider the radially symmetric semilinear Dirichlet problem \[ (1)\quad -\Delta u=f(| x|,u),\quad x\in B,\quad u|_{\partial B}=0, \] where B is the unit disc in \({\mathbb{R}}^ 2\) centred at the origin. Sufficient conditions on f are given for (1) to have a positive radial solution u in B. Theorems are obtained for superlinear nonlinearities f of subcritical, critical, and supercritical type. A constrained variational method is employed involving approximation of the energy functional by a functional satisfying the Palais-Smale condition.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 49J20
zbMATH DE Number: 4115120
zbMATH Keywords: semilinear
zbMATH Keywords: elliptic
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: positive radial solution
zbMATH Keywords: critical
zbMATH Keywords: superlinear
zbMATH Keywords: energy functional
zbMATH Keywords: Palais-Smale condition
reviewed by: Charles A. Swanson
MaRDI profile type: Publication
cites work: Q3359113
cites work: Ground states and Dirichlet problems for \(-\Delta u=f(u)\) in \(R^ 2\)
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Q3756181
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cites work: On a Class of Nonlinear Second-Order Differential Equations
full work available at URL: https://doi.org/10.1007/bf02874647
OpenAlex ID: W1977096260
description: scientific article; zbMATH DE number 4115120