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

Abstract

Abstract In this paper, we study the existence and nonexistence of positive solutions 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 $$\begin{aligned} -\Delta _p u = \frac{\lambda }{u^{\alpha }} + f(x,u,\nabla u) \end{aligned}$$ - Δ p u = λ u α + f ( x , u , ∇ u ) in a bounded, smooth domain $$\Omega $$ Ω . The convection term has exponents with no upper limitations neither in u nor in $$\nabla u$$ ∇ 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.