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 Duis an affine function; no conditions being placed on the behavior of the solution u.Bernstein's 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 BernsteinLiouville 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.

A remark on entire solutions of quasilinear elliptic equations

“…As in [1], we shall consider entire solutions of quasilinear elliptic equations div A (x, u, Du) = B(x, u, Du), (1.1) in the special strongly coercive case in which A (x, z, ρ) = A(x, z, ρ)ρ, B(x, z, ρ)z 0, (1.2) where A(x, z, ρ) is a positive, continuously differentiable scalar function, defined for x ∈ R n , z ∈ R \ {0}, ρ ∈ R n \ {0}, and where also A (x, z, 0) = 0, B(x, 0, 0) = 0. We assume moreover that the following "large radii conditions" of p-Laplace type are valid, that is for |x| R 0 , and all z = 0, ρ = 0,…”

“…With these assumptions, it is not hard to check that the principal conditions (3), (4), (8), (9) of [1] are valid for Eq. (1.1), and correspondingly that the results of [1] hold for (1.1).…”

“…With these assumptions, it is not hard to check that the principal conditions (3), (4), (8), (9) of [1] are valid for Eq. (1.1), and correspondingly that the results of [1] hold for (1.1). The remarkable nature of the main operational conditions (1.3)-(1.4) can hardly be overemphasized, especially that they are required to hold only for |x| R 0 , where R 0 can be arbitrarily large.…”

“…For model examples and general comments, as well as for comparison with important and closely related work, we refer to the introduction and bibliography of [1]. For simplicity here we shall consider only entire C 1 distribution solutions u of (1.1), that is, functions u of class C 1 (R n ) such that A (·, u, Du), B(·, u, Du) ∈ L 1 loc R n , R n A (x, u, Du) · Dϕ + B(x, u, Du)ϕ dx = 0 for all functions ϕ ∈ C 1 0 (R n ).…”

“…In this note, using the main comparison Theorem 3.6.4 of [2], we cover the open cases of Theorems B, D, E of [1] as described in Problems 2 and 3 of Section 12 of [1]. Moreover, we provide a significant improvement of Theorem C of [1], see Theorem 1.3.…”