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.

### An abstract three critical points theorem and applications

#### Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11586/68017
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