Positive solutions of quasi-linear elliptic equations with dependence on the gradient

In the present paper we prove a multiplicity theorem for a quasi-linear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary parametric problem. Other basic tools used in our paper are sub-supersolution techniques, Schaefer's fixed point theorem, regularity results and strong maximum principle. In our hypotheses we only require a general growth condition with respect to the solution and its gradient, and an assumption near zero involving the first eigenvalue of the negative -Laplacian operator.