An abstract three critical points theorem and applications

In this paper we state an abstract multiplicity theorem which generalizes the well known Pucci-Serrin result as it allows one to prove the existence of a third critical point for functionals which are smooth in a Banach space but satisfy a kind of Palais-Smale condition with respect to a weaker norm. This result applies for proving that, under suitable assumptions, the functional \[ J_\lambda(u) = \int_\Omega A(x,u)(|\nabla u|^p - \lambda |u|^p)dx + \int_\Omega G(x,u) dx \] admits at least three distinct critical points in the Banach space $W^{1,p}_0(\Omega) \cap L^\infty(\Omega)$ but if $\lambda$ is large enough