For the system of $d$-dim stochastic differential
equations,
dX^{\varepsilon} (t) = b(X^{\varepsilon}(t)) dt + \varepsilon dW(t),
\quad t \in [0, 1]
X^{\varepsilon} (0) = x^0 \in R^d
where $b$ is smooth except possibly along the hyperplane
$x_1 = 0$, we shall consider the large deviation principle for the lawof the
solution diffusion process and its occupation time as
$\varepsilon\rightarrow0$. In other words, we consider
$P(\|X^\varepsilon-\varphi\|<\delta,\|u^{\varepsilon}-\psi\|\<\delta)$
where $u^\varepsilon(t)$ and $\psi(t)$ are the occupation times of
$X^\varepsilon$ and $\varphi$ in the positive half space $\{x\in R^d: x_1>0\}$,
respectively. As a consequence, an unified approach of the lower level large
deviation principle for the law of $X^\varepsilon(\cdot)$ can be obtained.