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 $r\in (p-1,p^*-1)$.
Categories: math.AP