A mountain pass-type theorem for vector-valued functions

The mountain pass theorem for scalar functionals is a fundamental result of the minimax methods in variational analysis. In this work we extend this theorem to the class of C1 functions f : Rn ! Rm, where the image space is ordered by the nonnegative orthant Rm + . Under suitable geometrical assumptions, we prove the existence of a critical point of f and we localize this point as a solution of a minimax problem. We remark that the considered minimax problem consists of an inner vector maximization problem and of an outer setvalued minimization problem. To deal with the outer set-valued problem we use an ordering relation among subsets of Rm introduced by Kuroiwa. In order to prove our result, we develop an Ekeland-type principle for set-valued maps and we extensively use the notion of vector pseudogradient.