This paper introduces the concept of exceptional family for nonlinear varia
tional inequality problems. Among other things, we show that the nonexisten
ce of an exceptional family is a sufficient condition for the existence of
a solution to variational inequalities. This sufficient condition is weaker
than many known solution conditions and it is also necessary for pseudomon
otone variational inequalities. From the results in this paper, we believe
that the concept of exceptional families of variational inequalities provid
es a new powerful tool for the study of the existence theory for variationa
l inequalities.