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<p/> <p>We consider the perturbed half-linear Euler differential equation <inline-formula> <graphic file="1029-242X-2009-104043-i1.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-104043-i2.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-104043-i3.gif"/></inline-formula>, with the subcritical coefficient <inline-formula> <graphic file="1029-242X-2009-104043-i4.gif"/></inline-formula>. We establish a Bargmann-type necessary condition for the existence of a nontrivial solution of this equation with at least <inline-formula> <graphic file="1029-242X-2009-104043-i5.gif"/></inline-formula> zero points in <inline-formula> <graphic file="1029-242X-2009-104043-i6.gif"/></inline-formula>.</p>