Liouville theorem for monotone solutions of the Lane-Emden system in a half-space

Abstract

The Lane-Emden system is an important model case of Hamitonian elliptic systems. We here consider the zero-Dirichlet problem in a half-space for arbitrary powers $p,q\ge1$. We prove that any nonnegative classical solution that is nondecreasing in the normal direction is necessarily trivial. No bound at infinity of any kind is assumed and this extends the known scalar monotonicity theorem. We also obtain the same conclusion for the scalar equation $-Δu=f(u)$ when $f\in C^1([0,\infty))$ is nonnegative, $f(0)=0$, $f\not\equiv0$, and $f(s)/s$ is nondecreasing for $s>0$. This relaxes the convexity and $C^2$ regularity requirements on $f$ in previous works. To prove these results we introduce a new method, completely different from the normal convexity approach from previous works, which relies on a delicate combination of techniques and could be applied to more general systems. Our basic tool is a suitable average of the linear interpolant between the solution and its normal limit on each finite strip. We next use stability estimates, combined with an energy argument for the scalar equation, or a contraction argument relying on Bessel potentials and the Schur test for the system. Increasing the strip height then forces the solution to vanish.