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

Abstract

Let $p_i\ge 2$ and consider the following anisotropic $p$-Laplace equation $$-\sum _{i=1}^N\frac{\partial }{\partial x_i}\left(\right|\frac{\partial u}{\partial x_i}{|}^{p_i-2}\frac{\partial u}{\partial x_i})=g(x)f(u),u>0\text{ in }\hat{I}\text{c◯}.$$ 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 $\hat{I}\text{c◯}$ and nonexistence results for $f(u)=-e^{\frac{1}{u}}$ or $-(u^{-\hat{I}´}+u^{-{\hat{I}}^3}),$ $\hat{I}´,{\hat{I}}^3>0$ with $\hat{I}\text{c◯}={\mathbb{R}}^N$ respectively.

18 pages