On the uniqueness of solution for a class of semilinear elliptic problems
type: Item
instance of: scholarly article
title: On the uniqueness of solution for a class of semilinear elliptic problems
zbMATH Open document ID: 0635.35032
DOI: 10.1016/0022-247X(87)90302-7
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The Dirichlet problem \(-\Delta u=f(u)+\rho h(x)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) (\(\Omega\subset\mathbb R^ N \) bounded, \(\partial \Omega\) smooth) is considered, where \(f\in C^1(\mathbb R)\), \(h\in L_ 2(\Omega)\) and \(f_{\pm}=\lim_{s\to \pm \infty} f'(s)\) exist. The equivalent problem (2) \(-\Delta v=\rho^{-1}f(\rho v)+h\) in \(\Omega\), \(v=0\) on \(\partial \Omega\) is linked to the limiting problem (3) \(-\Delta v=f_+v\) \(--f_-v\) \(-+h\) in \(\Omega\), \(v=0\) on \(\partial \Omega\) by Theorem 1: If \(h(x)\neq 0\), (E) \(\| v\| \leq M\) for all solutions of (2) with \(\rho \geq \rho_ 0\) and (ND) any problem \(-\Delta w=(f_+\chi_ A+f_-\chi_ B)w\) in \(\Omega\), \(w=0\) on \(\partial \Omega\) \((A\cup B=\Omega\), \(A\) and \(B\) have positive measure) has only the zero solution, then uniqueness for problem (3) implies uniqueness for problem (2). Several sufficient conditions for (E), (ND) and the unique solvability of (3) are given in terms of the eigenvalues \(\lambda_ 1<\lambda_ 2<\lambda_ 3<...\) are the corresponding eigenspaces \(N_ i\). Theorem 2 states that the assumptions \(f_+,f_-\leq \lambda_ 1\) or \(\lambda_ k\leq f_+\), \(f_-\leq \lambda_{k+1}\) (but not \(f_+=f_-=\lambda_ i)\), \(h\in L^p(\Omega)\) for some \(p>N\) plus a non-orthogonality assumption in case \(f_-\) or \(f_+\) coincides with an eigenvalue imply that (3) has a unique solution and (ND) holds.
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 4034214
zbMATH Keywords: semilinear elliptic problems
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: uniqueness
zbMATH Keywords: eigenvalues
author: David G. Costa
author: Djairo G. de Figueiredo
author: José Valdo A. Gonçalves
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4034214