Existence results for some anisotropic singular problems via the sub-supersolution method

Abstract

Using the sub-super solution method, we prove the existence of the solutions for the following anisotropic problem with singularity:
\[\begin{cases}
-\sum\limits_{i=1}^{N} \partial_{i} \left({| \partial_{i} u \vert}^{p_{i}-2} \partial_{i} u \right) = f(x,u) &\qquad\text{in $\;\;\Omega$,}\\
u>0 &\qquad\text{in $\;\;\Omega $,}\\
u=0 &\qquad\text{on $\;\;\partial\Omega $,}
\end{cases}\]

where \(\Omega \subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary and a given singular nonlinearity \(f:\Omega\times(0,\infty)\longrightarrow [0,\infty)\).