Summary: Let \(\Omega\) be a bounded open set of \(\mathbb{R}^N\), \(N\geq 1\). We look for solutions of the quasilinear Dirichlet problem \[ u\in H^1_0(\Omega),\quad -\text{div}(A(x,u)Du)= g(x,u), \] where \(A(x,s)\) is a Carathéodory elliptic matrix and \(g(x,s)\) is a Carathéodory function increasing with respect to \(s\). We adapt the classical method of sup-super-solutions in order to obtain the existence of a solution \(u\). Then we apply that result to prove the existence of positive (non-trivial) solutions of some quasilinear elliptic boundary value problems.