Monotonicity and one-dimensional symmetry for solutions of -Delta(p)u = f(u) in half-spaces

We prove a weak comparison principle in narrow domains for sub-super solutions to -Delta(p)u = f(u) in the case 1 < p <= 2 and f locally Lipschitz continuous. We exploit it to get the monotonicity of positive solutions to -Delta(p)u = f(u) in half spaces, in the case 2N+2/N+ 2 < p <= 2 and f positive. Also we use the monotonicity result to deduce some Liouville-type theorems. We then consider a class of sign-changing nonlinearities and prove a monotonicity and a one-dimensional symmetry result, via the same techniques and some general a-priori estimates.