  In this paper, we mainly discuss the local regularity of the solution to the
following problem \begin{align*} \begin{cases} -\dive({\bf{A}}(x)\nabla
u(x))=f(x),&~x\in\Omega,\\ u(x)=0,&~x\in\partial\Omega, \end{cases}
\end{align*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$. In
particular, we are concerned with the connection between the regularity of the
solution $u$ and the integrability of the coefficient matrix ${\bf{A}}(x)$ as
well as the nonhomogeneous term $f$. To be more precise, our first result is to
prove that the maximum norm of $u$ can be controlled by $\|f\|_{s}$ with $f\in
L^s(\Omega),~s>\frac{nq}{2q-n},~q>\frac{n}{2}$. Meanwhile, we construct some
counterexamples to illustrate the index $\frac{nq}{2q-n}$ being sharp.
Subsequently, we give an improved upper bound for the maximum norm of $u$.
Namely, there exists a positive constant $C$ such that $$\|u\|_{\infty}\leq
C\|f\|_{\frac{nq}{2q-n}}\left[\log\left(\frac{\|f\|_{s}}{~~~~~\|f\|_{\frac{nq}{2q-n}}}+1\right)+1\right].$$
Specially, the main difference of our approach compared to the arguments of
[\ref{CUR}, \ref{XU}] is to construct two classes of truncation functions to
remove the assumption of the boundedness of $u$. Finally, based on the previous
results and Moser iteration argument, we derive the Harnack inequality of $u$
from which the H\"older continuity of the solution follows. In addition, we
also find that the Lebesgue space $L^{\frac{n}{2}}(\Omega)$ to which the
inverse of the smallest eigenvalue $\lambda(x) $ of the matrix {\bf{A}}(x)
belongs is essentially sharp in order to establish local boundedness and the
H\"older continuity of the solution.
