For a wide class of nonlinearities f(u) satisfyingbut not necessarily Lipschitz continuous, we study the quasi-linear equationwhere T = {x = (x1, x2, …, xN) ∈ RN: x1 > 0} with N ≥ 2. By using a new approach based on the weak maximum principle, we show that any positive solution on T must be a function of x1 only. Under our assumptions, the strong maximum principle does not hold in general and the solution may develop a flat core; our symmetry result allows an easy and precise determination of the flat core.