Liouville type theorems for anisotropic degenerate elliptic equations on strips

We establish ($L^\infty$) Liouville type theorems for anisotropic degenerate elliptic equations in divergence form on the strip $S=\R^{N-1}\times (-1,1)$ where $x=(x',\lambda)$. The model equation is $div_{x'} (w_1 \nabla_{x'}\sigma)+\partial_\lambda (w_1w_2 \partial_\lambda \sigma)=0$, where $w_i(x',\lambda)$ are positive and locally bounded in $S$. We deduce them from an oscillation decrease argument under appropriate conditions on the weight functions $w_i$; the key one being the existence of a positive unbounded supersolution (or ''almost-supersolution") close to the degeneration set $\partial S$. For example our approach works in the case $w_1=1-|\lambda|$ and $w_2=(1-|\lambda|)^2$, for which the corresponding ($L^\infty$) Liouville type theorem entails an alternative proof of the (known) positive answer to a famous conjecture of De Giorgi in any space dimension under the additional assumption that the zero level set of the solution is a Lipschitz graph; moreover it is related to an ($L^\infty$) Liouville type theorem for an isotropic degenerate elliptic equation on $\mathbb R^N$ with a one-dimensional weight decaying exponentially at infinity in one direction.