  Resonant inelastic x-ray scattering (RIXS) at the copper K absorption edge
has been performed for heavily overdoped samples of La$_{2-x}$Sr$_{x}$CuO$_{4}$
with $x= 0.25$ and 0.30. We have observed the charge transfer and
molecular-orbital excitations which exhibit resonances at incident energies of
$E_i= 8.992$ and 8.998 keV, respectively. From a comparison with previous
results on undoped and optimally-doped samples, we determine that the
charge-transfer excitation energy increases monotonically as doping increases.
In addition, the $E_i$-dependences of the RIXS spectral weight and absorption
spectrum exhibit no clear peak at $E_i = 8.998$ keV in contrast to results in
the underdoped samples. The low-energy ($\leq 3$ eV) continuum excitation
intensity has been studied utilizing the high energy resolution of 0.13 eV
(FWHM). A comparison of the RIXS profiles at $(\pi ~0)$ and $(\pi ~\pi)$
indicates that the continuum intensity exists even at $(\pi ~\pi)$ in the
overdoped samples, whereas it has been reported only at $(0 ~0)$ and $(\pi ~0)$
for the $x=0.17$ sample. Furthermore, we also found an additional excitation on
top of the continuum intensity at the $(\pi ~\pi)$ and $(\pi ~0)$ positions.
