Let [Formula: see text] be a polynomial ring in [Formula: see text] indeterminates with coefficients in the field [Formula: see text] of characteristic [Formula: see text] and [Formula: see text] be the ring of differential operators over [Formula: see text]. In this paper, we introduce the notion of generalized Eulerian [Formula: see text]-modules for characteristic [Formula: see text] and establish their properties. We show that if [Formula: see text] is any graded Lyubeznik functor on the category of modules over [Formula: see text] then [Formula: see text] is a generalized Eulerian [Formula: see text]-module. As a consequence, we prove that all socle elements of module [Formula: see text] are concentrated in degree [Formula: see text] where [Formula: see text] is an irrelevant maximal ideal of [Formula: see text].

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