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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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