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On the uniqueness of positive solutions of the Dirichlet problem \(-\Delta u=\lambda \,\sin \,u\). (English) Zbl 0569.35037
Nonlinear partial differential equations and their applications, Coll. de France Semin., Vol. VII, Paris 1983-84, Res. Notes Math. 122, 80-83 (1985).
[For the entire collection see Zbl 0559.00005.]
The author isolates a class of bounded, smoothly bounded domains \(\Omega\) in \({\mathbb{R}}^ N\) (N\(\geq 2)\) for which the problem \(-\Delta u=f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) has a unique positive solution when f has a certain oscillatory property; for example, when \(f(u)=\lambda \sin u\).
Reviewer: D.E.Edmunds

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs