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

Let $p_i\ge 2$ and consider the following anisotropic $p$-Laplace equation $$-\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(\mid \frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}{\mid }^{p_i-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i})=g(x)f(u),u>0\text{ in }\mathrm{\Omega }\mathrm{.}$$ 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 $\mathrm{\Omega }$ and nonexistence results for $f(u)=-e^{\frac{1}{u}}$ or $-(u^{-\delta }+u^{-\gamma }),$ $\delta ,\gamma >0$ with $\mathrm{\Omega }={\mathbb{R}}^N$ respectively.