A value function and applications to translation invariant semilinear elliptic equations on unbounded domains
type: Item
instance of: scholarly article
title: A value function and applications to translation invariant semilinear elliptic equations on unbounded domains
zbMATH Open document ID: 0820.35106
author: Ian Schindler
published in: Differential and Integral Equations
publication date: 1995-02-01T00:00:00Z
review text: The nonlinear elliptic boundary value problem \(- \Delta u + u = \lambda f(x,u)\) in \(G\), \(u = 0\) on \(\partial G\), \(\lim_{| x | \to \infty} u(x) = 0\) is studied in this paper. Here \(G\) is an unbounded cylindrical domain, \(f\) is a continuous function with periodic dependence on \(x\) in the unbounded direction and such that \[ \lim_{s \to 0} {f(x,s) \over s} = 0, \quad \lim_{| s | \to \infty} {f(x,s) \over s^ r} = 0, \] where \(r = {N + 2 \over N - 2}\) \((N\) is the space dimension). In particular, it is proved that there is a weak solution (in the Sobolev space \(H^ 1_ 0 (G))\) of this problem for a dense set of values of the real parameter \(\lambda\). Moreover, if \(G\) contains a plane and \(f\) is autonomous, then there is a solution if \(\lambda\) is large enough. Proofs are given by combining properties of critical value functions and the concentration-compactness method of P. L. Lions. Existence for \(\lambda\) large when \(G\) contains a plane follows from a result of O. Kavian.
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 47J10
zbMATH DE Number: 716355
zbMATH Keywords: critical value function
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: unbounded domain
zbMATH Keywords: concentration-compactness method
reviewed by: Jesús Hernández
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 716355