  The exact solution of N- dimensional radial Schr\"odinger equation with the
generalized Cornell potential has been obtained using the Laplace
transformation (LT) method. The energy eigenvalues and the corresponding wave
functions for any state have been determined. The eigenvalues for some special
cases of the generalized Cornell potential are obtained. The present results
are applied to calculate the mass spectra of heavy quarkonium systems such as
charmonium and bottomonium and the bc meson. A comparison is discussed with the
experimental data and recent works. The present results are improved in
comparison with other recent studies and are in a good agreement with the
experimental data. The effect of the dimensional number (N) on the meson masses
has been studied. We note that the meson masses increase in higher dimensions.
