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On an anisotropic p-Laplace equation with variable singular exponent (2104.03858v2)
Published 8 Apr 2021 in math.AP
Abstract: 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 $g(u)=ur$ for $r\in (p-1,p*-1)$.