Accepted Paper

Inserted: 12 may 2011

Journal: Calc. Var. Partial Differential Equations
Year: 2010

Abstract:

We prove a weak comparison principle in narrow domains for sub-super solutions to $-\Delta_p u=f(u)$ in the case $1<p \le 2$ 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 $\frac{2N+2}{N+2}<p \le 2$ 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.

Download:

• Monotonicity_FMS-rev.pdf