# Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

## Abstract

In 1978 De Giorgi formulated the following conjecture. *Let be a solution of in all of such that and in Is it true that all level sets . of are hyperplanes, at least if *? Equivalently, does depend only on one variable? When this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for , The question, however, remains open for . The results for . and 3 apply also to the equation for a large class of nonlinearities .

## 1. Introduction

This paper is concerned with the study of bounded solutions of semilinear elliptic equations in the whole space under the assumption that , is monotone in one direction, say, in The goal is to establish the one-dimensional character or symmetry of . namely, that , only depends on one variable or, equivalently, that the level sets of are hyperplanes. This type of symmetry question was raised by De Giorgi in 1978, who made the following conjecture – we quote (3), page 175 of Reference DG:

When this conjecture was recently proved by Ghoussoub and Gui ,Reference GG. In the present paper we prove it for The conjecture, however, remains open in all dimensions . The proofs for . and use some techniques in the linear theory developed by Berestycki, Caffarelli and Nirenberg Reference BCN in one of their papers on qualitative properties of solutions of semilinear elliptic equations.

The question of De Giorgi is also connected with the theories of minimal hypersurfaces and phase transitions. As we explain later in the introduction, the conjecture is sometimes referred to as “the of the Bernstein problem for minimal graphs”. This relation with the Bernstein problem is probably the reason why De Giorgi states “at least if -version in the above quotation. ”

Most articles dealing with the question of De Giorgi have also considered the conjecture in a slightly simpler version. It consists of assuming that, in addition,

Here, the limits are *not* assumed to be uniform in

The positive answers to the conjecture for

The following are our results for

Note that the direction

Theorem 1.1 applies to

for some

The hypothesis (Equation 1.4) made on

The following result establishes for

Our proof of Theorem 1.1 will only require

The first partial result on the question of De Giorgi was found in 1980 by Modica and Mortola Reference MM2. They gave a positive answer to the conjecture for

In 1985, Modica Reference M1 proved that if

In 1994, Caffarelli, Garofalo and Segala Reference CGS generalized this bound to more general equations. They also showed that, if equality occurs in (Equation 1.6) at some point of

Under the additional assumption that

Using a one-dimensional arrangement argument, Farina Reference F1 proved the conclusion

Our proof of the conjecture of De Giorgi in dimension

The energy functional in

has

and

is a nondecreasing function of

Note that the estimate of Theorem 1.3 is clearly true assuming that

Finally, we recall the heuristic argument that connects the conjecture of De Giorgi with the Bernstein problem for minimal graphs. For simplicity let us suppose that

and the penalized energy of

Note that

In a forthcoming work Reference AAC with Alberti, we will use new variational methods to study the conjecture of De Giorgi in higher dimensions.

In section 2 we prove Theorems 1.1 and 1.3. Section 3 is devoted to establishing Theorem 1.2.

## 2. Proof of Theorem 1.1

To prove the conjecture of De Giorgi in dimension 3, we will use the energy estimate of Theorem 1.3. It is this estimate that will allow us to apply, when

The study of this type of Liouville property, its connections with the spectrum of linear Schrödinger operators, as well as its applications to symmetry properties of solutions of nonlinear elliptic equations, were developed by Berestycki, Caffarelli and Nirenberg Reference BCN. In the papers Reference BCN and Reference GG, this Liouville property was shown to hold under various decay assumptions on

Before proving Theorem 1.3 and Proposition 2.1, we use these results to give the detailed proof of Theorem 1.1. First, we establish some simple bounds and regularity results for the solution

Indeed, applying interior

with

Next, we verify that

in particular, we have that

Indeed, since

in the weak sense, for every index

When carried out in dimension 2, the previous proof is essentially the one given in Reference GG to establish their extended version of the conjecture of De Giorgi for

We turn now to the