  The paper contains lower bounds on the counting function of the positive
eigenvalues of the interior transmission problem when the latter is elliptic.
In particular, these bounds justify the existence of an infinite set of
interior transmission eigenvalues and provide asymptotic estimates from above
on the counting function for the large values of the wave number. They also
lead to certain important upper estimates on the first few interior
transmission eigenvalues. We consider the classical transmission problem as
well as the case when the inhomogeneous medium contains an obstacle.
