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)\left\{\begin{split} -Δ_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }Ω,\\ u&>0\text{ in }Ω,\,u=0\text{ on }\partialΩ, \end{split}\right. \end{equation*} under the assumption $\hat{I}\text{c◯}$ is a bounded smooth domain in ${\mathbb{R}}^N$ with $p,N\ge 2$, $\la>0$ 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 $r\in (p-1,p^{\ast }-1)$.

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