  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.
