Positive solutions of singular elliptic problems with unbounded exponents and unbounded convection term

In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-{\mathrm{\Delta }}_{p}u=\frac{\lambda }{u^{\alpha }}+f(x,u,\mathrm{\nabla }u)$$ in a bounded, smooth domain $\mathrm{\Omega }$. The convection term has exponents with no upper limitations neither in $u$ nor in $\mathrm{\nabla }u$. This is somewhat unexpected and rare. So, we address a wide range of problems not yet contained in the literature. The solution of the problem combines the definition of an auxiliary problem, the method of sub- and super-solution and Schauder’s fixed point theorem.