Using monotonicity methods, the Lagrange multiplier rule and some
variational arguments, we consider a type of localization results
pertaining to the existence of critical points to action functionals on
a closed ball. A variant of the Schechter critical point theorem on a
ball in Hilbert and Banach spaces is obtained. Applications to nonlinear
Dirichlet problem and to partial difference equations are given in the
final part of this paper.