An exact multiplicity result for a class of symmetric problems
Summary: We consider positive solutions of a class of semilinear problems \[ u'' + \lambda a(x)f(u) = 0, \quad - 1 < x < 1, \quad u(-1) = u(1) = 0, \] with even and positive \(a(x)\), depending on a positive parameter \(\lambda \). In case \(f(u)\) is convex, an exact multiplicity result was given in our paper with \textit{Y. Li} and \textit{T. Ouyang} [Proc. R. Soc. Edinb., Sect. A, Math. 126, No. 3, 599--616 (1996; Zbl 0855.34022)]. It was observed in our paper with \textit{J. Shi} [Electron. J. Differ. Equ. 2000, 311--322 (2000; Zbl 0970.34015)] that convexity requirement can be relaxed for large \(u\). We show that convexity requirement can also be relaxed for small \(u\).
scientific article; zbMATH DE number 5296710