In terms of the upper bounds of a second-order elliptic operator acting on specific Lyapunov-type functions with compact level sets, sufficient conditions are presented for the corresponding Dirichlet form to satisfy the Poincaré and the super-Poincaré inequalities. Here, the elliptic operator is assumed to be symmetric on $L^2(\mu)$ with some probability measure $\mu$. As applications, proofs are given for a class of (non-symmetric) diffusion operators generating $C_0$-semigroups on $L^1(\mu)$: that their $L^p(\mu)$-essential spectrum is empty for $p>1$. This follows since it is proved that their $C_0$-semigroups are compact.Supported in part by the NNSFC for Distinguished Young Scholars (10025105), the 973-Project and TRAPOYT in China.