Higher-dimensional solutions for a nonuniformly elliptic equation

We prove $m$-dimensional symmetry results, that we call $m$-Liouville theorems, for stable and monotone solutions of the following nonuniformly elliptic equation \begin{eqnarray*}\label{mainequ} - div(\gamma(\mathbf x') \nabla u(\mathbf x)) =\lambda (\mathbf x' ) f(u(\mathbf x)) \ \ \text{for}\ \ \mathbf x=(\mathbf x',\mathbf x'')\in\mathbf{R}^d\times\mathbf{R}^{s}=\mathbf{R}^n, \end{eqnarray*} where $0\le m<n$ and $0<\lambda,\gamma$ are smooth functions and $f\in C^1(\mathbf R)$. The interesting fact is that the decay assumptions on the weight function $\gamma(\mathbf x')$ play the fundamental role in deriving $m$-Liouville theorems. We show that under certain assumptions on the sign of the nonlinearity $f$, the above equation satisfies a 0-Liouville theorem. More importantly, we prove that for the double-well potential nonlinearities, i.e. $f(u)=u-u^3$, the above equation satisfies a $(d+1)$-Liouville theorem. This can be considered as a higher dimensional counterpart of the celebrated conjecture of De Giorgi for the Allen-Cahn equation. The remarkable phenomenon is that the $\tanh$ function that is the profile of monotone and bounded solutions of the Allen-Cahn equation appears towards constructing higher dimensional Liouville theorems.