A non-local elliptic equation of Kirchhoff-type
for
with Dirichlet boundary conditions is investigated for the cases where
. It is well known that with the non-local effect removed and
, a branch of positive solutions bifurcates from infinity at
and no positive solution exists whenever
for some
(see K. J. Brown, Calc. Var.
22, 483-494, 2005),where
is the principal eigenvalue of the linear problem
. As a consequence of the non-local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for
. Moreover, regions with three positive solutions are found for small value of
. Comparisons are also made of the results here with those of the elliptic problem in the absence of the non-local term under the same prescribed conditions using numerical simulations.