  Langmuir waves (LWs), which are believed to play a crucial role in the plasma
emission of solar radio bursts, can be excited by streaming instability of
energetic electron beams. However, solar hard X-ray observations imply that the
energetic flare electrons usually have a power-law energy distribution with a
lower energy cutoff. In this paper, we investigate LWs driven by the power-law
electrons. The results show that power-law electrons with the steepness cutoff
behavior can excite LWs effectively because of the population inversion
distribution below the cutoff energy ($E_c$). The growth rate of LWs increases
with the steepness index ($\delta$) and decreases with the power-law index
($\alpha$). The wave number of the fastest growing LWs ($k\lambda_D$),
decreases with the characteristic velocity of the power-law electrons
($v_{c}=\sqrt{2E_{c}/m_{e}}$) and increases with the thermal velocity of
ambient electrons ($v_T$). This can be helpful for us to understand better the
physics of LWs and the dynamics of energetic electron beams in space and
astrophysical plasmas.
