On an anisotropic p-Laplace equation with variable singular exponent

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 $\mathrm{\Omega }$ is a bounded smooth domain in ${\mathbb{R}}^N$ with $p,N\ge 2$, $\text{la}>0$ and $0<q\in C(\overline{\text{Om}})$. 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 $r\in (p-1,p^{\ast }-1)$.