We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $-{\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)+b\left(x\right)u\left(x\right)+div\left(\Phi \left(u\left(x\right)\right)\right)=g\left(x\right)-{\sum }_{j=1}^n{f}_j\left(x\right)$ on Ω in the setting of the space H₀(Ω).

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...