We consider a C¹-functional ψ defined on the "Neumann" Sobolev space W_n^1,p(Ω). If M is a C¹-submanifold, then for ψ|_M we show that any local C_n¹($\overline{\Omega}$)-minimizer is also a local W_n^1,p(Ω)-minimizer. Then we use this general result on local minimizers to show that a nonlinear parametric Neumann problem driven by the p-Laplace differential operator and restricted on a sphere, has at least three distinct smooth solutions.