*Accepted Paper*

**Inserted:** 16 sep 2010

**Journal:** Journal of Differential Equations

**Year:** 2010

**Abstract:**

\noindent**Abstract.** {\it A famous theorem of Sergei Bernstein
says that every entire solution $u = u(x)$, $x \in *R*^{2}$ of
the minimal surface equation
$$
{\rm div}\,\left\{ {Du \over \sqrt{1 + Du^{{2}}}} \right\} = 0
$$
is an affine function; no conditions being placed on the behavior of
the solution $u$.

Bernstein's Theorem continue to hold up to dimension $ n = 7$ while it fails to be true in higher dimensions, in fact if $x \in *R*^{n}$, with $n \geq 8$, there exist entire non-affine minimal graphs (Bombieri, De Giorgi and Giusti).

Our purpose is to consider an extensive family of quasilinear elliptic-type equations which has the following strong Bernstein-Liouville property, that $u \equiv 0$ for {\it any} entire solution $u$, no conditions whatsoever being placed on the behavior of the solution (outside of appropriate regularity assumptions). In many cases, moreover, no conditions need be placed even on the dimension $n$. We also study the behavior of solutions when the parameters of the problem do not allow the Bernstein--Liouville property, and give a number of counterexamples showing that the results of the paper are in many cases best possible.}

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