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 valueproblem involving the p-biharmonic          0, on( ) 1 ( , ) , in , 2*2u uf x u u upu u p q where  is a bounded domain in rn with smooth boundary  . we prove that the problem has atleast two nontrivial nonnegtive solutions when the parameter  belongs to a certain subset of r .