On an Anisotropic $p$-Laplace equation with variable singular exponent

Abstract

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P) \begin{cases} -\Delta_{H,p}u =\frac{\lambda f(x)}{u^{q(x)}}+g(u) \ \text{ in }\Omega,\\ u > 0 \text{ in }\Omega,\ \ \ \ u=0\text{ on }\partial\Omega, \end{cases} \end{equation*} under the assumption $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\lambda>0$ and $0 < q \in C(\bar \Omega)$. 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 $r\in (p-1,p^*-1)$.