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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

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