In 1978 De Giogi conjectured that if $u(x)$, $x=(x_1, \dots, x_n) \in R^n$ is a global bounded solution of the non-linear PDE $$\Delta u + u - u^3 =0, \eqno(1)$$ which is monotone in the $x_1$ direction, then $u$ is essentially one-dimensional. Recently Ghoussoub and Gui have proved this conjecture for $n=2$.

It turns out that this problem can be reformulated in terms of a Liouville property for a divergence form operator $${\cal L} = \nabla \sigma^2 \nabla. \eqno(2)$$ Here $\sigma(x)$, $x \in R^n$ is a positive (but not uniformly positive) function, defined in terms of the solution $u$. This problem can be studied using probabilistic techniques.

In this talk I will describe recent work (with R.F. Bass and C. Gui) in which we prove a Liouville theorem for some operators of the form (2). As a consequence, we can prove the conjecture under some additional hypotheses.