A mountain pass theorem for a suitable class of functions
In this article, a mountain pass theorem is obtained for a functional of the form \(J=\Phi-\Psi\) defined on a reflexive real Banach space, where \(\Phi\) and \(\Psi\) are continuously Gâteau differentiable functions, \(\Phi\) is convex, and \(J_M = \Phi-\Psi_M\) satisfies the Palais--Smale condition (PS)\(_c\) with \(\Psi_M\) the cut-off function at a suitable constant \(M\). This mountain pass theorem is combined to a result on the existence and localization of two local minima of \(J\) [\textit{G.\,Bonanno}, J.~Math.\ Anal.\ Appl.\ 299, No.\,2, 600--614 (2004; Zbl 1071.34015)] to deduce a three critical points result without assuming the coercivity of \(J\). This result is then applied to obtain the existence of three nonnegative solutions of two-point boundary value problems with no asymptotic conditions.
scientific article; zbMATH DE number 5566197
scientific article