This paper is concerned with a class of nonlinear elliptic equations with a generalized Hardy potential

$$-\mathrm{\Delta }u=\frac{\lambda }{|x{|}^{\alpha }}u-b(x)g(u),x\in \mathrm{\Omega }\mathrm{\setminus }\{0\},$$

where $\alpha >0$, $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 3)$ is a bounded smooth domain containing the origin. We establish the existence, nonexistence, and asymptotic behavior of positive solutions. When the potential function $|x|{}^{-\alpha }$ has strong singularity at the origin, we obtain some qualitative properties of positive solutions.