On an anisotropic problem with singular nonlinearity having variable exponent

Abstract

We consider the following anisotropic problem, with singular nonlinearity having a variable exponent $$\begin{aligned} \left\{ \begin{array}{ll} -\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =\frac{f}{u^{\gamma (x) }} &{} \quad in~\Omega , \\ u=0 &{} \quad on~\Omega , \\ u\ge 0 &{} \quad in~\Omega ; \end{array} \right. \end{aligned}$$ where \(\Omega \) is a bounded regular domain in \({\mathbb {R}}^{N}\) and \(\gamma (x)>0\) is a smooth function, having a convenient behavior near \(\partial \Omega .\)f is assumed to be a non negative function belonging to a suitable Lebesgue space \(L^{m}\left( \Omega \right) .\) We will also assume without loss of generality that \(2\le p_{1}\le p_{2}\le \cdots \le p_{N}.\) Using approximation techniques, we obtain existence and regularity of positive solutions to the considered problem.