We show the $L^r(\mathbb{R}^d, μ)$-uniqueness for any $r \in (1, 2]$ and the essential self-adjointness of a Dirichlet operator $Lf = Δf +\langle \frac{1}ρ\nabla ρ, \nabla f \rangle$, $f \in C_0^{\infty}(\mathbb{R}^d)$ with $d \geq 3$ and $μ=ρdx$. In particular, $\nabla ρ$ is allowed to be in $L^d_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ or in $L^{2+\varepsilon}_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ for some $\varepsilon>0$, while $ρ$ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.