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Uniqueness of the solution of a semilinear boundary value problem. (English) Zbl 0576.35044
Der Autor zeigt die Eindeutigkeit der positiven Lösung des Randwertproblems \[ \epsilon^ 2\Delta u+f(x,u)=0,\quad x\in \Omega,\quad u(x)=0,\quad x\in \partial \Omega \] für \(0<\epsilon \ll 1\), wobei die Nichtlinearität f die Bedingungen (i) f(x,0)\(\geq 0\), \(x\in {\bar \Omega}\); (ii) \(f_ u(x,0)>0\), wenn \(f(x,0)=0\); (iii) Es gibt \(a\in C^ 1({\bar \Omega})\), so daß \(f(x,u)>0\) für \(0<u<a(x)\) und \(f(x,u)<0\) für \(u>a(x)\); (iv) \(f_ u(x,a(x))<0\), \(x\in {\bar \Omega}\) erfüllt und wobei f und der Rand des Gebietes \(\Omega \subset {\mathbb{R}}^ n\) genügend glatt sind.
Reviewer: W.Wendt

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B50 Maximum principles in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:
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