Existence of three solutions for a perturbed two-point boundary value problem (Q975285)

This paper deals with the existence of three solutions for the following perturbed two-point boundary value problem \[ -u''=\lambda f(u)+\mu g(x,\mu), \quad x\in [0,1], \] \[ u(0)=u(1)=0, \] where \(\lambda\) and \(\mu\) are positive real parameters, and \(f\) and \(g\) are continuous functions. The authors establish precise values of \(\lambda\) and \(\mu\) for which the above problem admits at least three classical solutions, which is in part an improvement of the work in [\textit{F. Cammaroto, A. Chinni} and \textit{B. Di Bella}, J. Math. Anal. Appl. 323, No.~1, 530--534 (2006; Zbl 1112.34307)]. The proof of the main result (Themorem 3.1) is based on two kinds of three-critical-point theorems obtained in [\textit{G. Bonanno} and \textit{P. Candito}, J. Differ. Equations 244, No.~12, 3031--3059 (2008; Zbl 1149.49007); \textit{G. Bonanno} and \textit{S. A. Marano}, Appl. Anal. 89, No.~1, 1--10 (2010; Zbl 1194.58008)].