Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions

In this note we present some uniqueness and comparison results for a class of problem of the form(0.1) − L u = H ( x , u , ∇ u ) + h ( x ) , u ∈ H 0 1 ( Ω ) ∩ L ∞ ( Ω ) , where Ω ⊂ R N , N ≥ 2 is a bounded domain, L is a general elliptic second order linear operator with bounded coefficients and H is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp.