Entire solutions of completely coercive quasilinear elliptic equations

Abstract

A famous theorem of Sergei Bernstein says that every entire solution u = u ( x ) , x ∈ R 2 , of the minimal surface equation div { D u 1 + | D u | 2 } = 0 is an affine function; no conditions being placed on the behavior of the solution u. Bernsteinʼs Theorem continues to hold up to dimension n = 7 while it fails to be true in higher dimensions, in fact if x ∈ R n , with n ⩾ 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 ≡ 0 for 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.