Existence and Nonexistence results for anisotropic p-Laplace equation with singular nonlinearities

Abstract

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }\Omega. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\frac{1}{u}$ in a bounded smooth domain $\Omega$ and nonexistence results for $f(u)=-e^\frac{1}{u}$ or $-(u^{-\delta}+u^{-\gamma}),$ $\delta,\gamma>0$ with $\Omega=\mathbb{R}^N$ respectively.