On $$p(\cdot )$$-Laplacian problem with singular nonlinearity having variable exponent

This paper deals with the existence and regularity of solutions for $$p(\cdot )$$ -Laplacian problem with singular nonlinearity having variable exponent. The model example is $$\begin{aligned} \left\{ \begin{array}{ll}{} -\Delta _{p(x)}u=\frac{f(x)}{u^{\alpha (x)}}, &{} x \in \Omega ,\\ u=0, &{} x \in \partial \Omega , \end{array}\right. \end{aligned}$$ where $$\alpha (x)>0$$ and f is a nonnegative function belonging to the Lebesgue space $$L^{m}(\Omega )$$ for some suitable $$m \ge 1$$ . The results show the dependence of the summability of f in some Lebesgue spaces and on the values of $$\alpha (x)$$ .