The Nehari manifold for a Navier boundary value problem involving the p-biharmonic

In this paper, we study the Nehari manifold and its application on the following Navier boundary value problem involving the p-biharmonic ?? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? 0, on ( ) 1 ( , ) , in , 2 * 2 u u f x u u u p u u p q ? where ? is a bounded domain in RN with smooth boundary ? ?. We prove that the problem has at least two nontrivial nonnegtive solutions when the parameter ? belongs to a certain subset of R .