  In this article, we study the following anisotropic p-Laplacian equation with
variable exponent given by \begin{equation*} (P)\left\{\begin{split}
-\Delta_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }\Omega,\\ u&>0\text{
in }\Omega,\,u=0\text{ on }\partial\Omega, \end{split}\right. \end{equation*}
under the assumption $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with
$p,N\geq 2$, $\la>0$ and $0<q \in C(\bar \Om)$. For the purely singular case
that is $g\equiv 0$, we proved existence and uniqueness of solution. We also
demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$
and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.
