  A celebrated conjecture due to De Giorgi states that any bounded solution of
the equation $\Delta u + (1-u^2) u = 0 \hbox{in} \R^N $ with $\pp_{y_N}u >0$
must be such that its level sets $\{u=\la\}$ are all hyperplanes, {\em \bf at
least} for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been
believed to exist. Based on a minimal graph $\Gamma$ which is not a hyperplane,
found by Bombieri, De Giorgi and Giusti in $\R^N$, $N\ge 9$, we prove that for
any small $\alpha >0$ there is a bounded solution $u_\alpha(y)$ with
$\pp_{y_N}u_\alpha >0$, which resembles $ \tanh (\frac t{\sqrt{2}}) $, where
$t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph
$\Gamma_\alpha := \alpha^{-1}\Gamma$. This solution constitutes a
counterexample to De Giorgi conjecture for $N\ge 9$.
