In this paper we study isometrically immersed hypersurfaces of the Euclidean space E^{n+1 satisfying the condition L_r H_(r+1)=\lambda H_(r+1 {for an integer r , where H_{r+1} is the (r+1) th mean curvature of hypersurface arising from its normal variations. Having assumed that on a hypersurface x:M^n ----> E^n+1 , the vector field H_r+1 be an eigenvector of the operator L_r with a constant real eigenvalue \lambda we show that , M^n has to be an L_r-biharmonic , L_r-1-type or L_r-null-2-type hypersurfaces. As an interesting result, we have that, the L_r-biharmonicity condition on the weakly convex Euclidean hypersurfaces implies the r-minimality.
