Calculus of Variations and Geometric Measure Theory
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A. Farina - L. Montoro - B. Sciunzi

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

created by farina on 12 May 2011


Accepted Paper

Inserted: 12 may 2011

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


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.


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