  This paper introduces a novel higher order Adams inequality that incorporates
an exact growth condition for a class of weighted Sobolev spaces. Our rigorous
proof confirms the validity of this inequality and provides insights into the
optimal nature of the critical constant and the exponent within the
denominator. Furthermore, we apply this inequality to study a class of ordinary
differential equations (ODEs), where we successfully derive both a concept of
the weak solution and a comprehensive regularity theory.
