Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient

The existence of positive solutions for nonlinear elliptic problems under Dirichlet boundary condition is studied as well as the compactness and directness of the solution set. The main novelties consist in the presence of a Leray–Lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution. Our approach relies on the method of sub-supersolution, nonlinear regularity theory and strong maximum principle.