Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$

Abstract

We study the existence and instability properties of saddleshaped solutions of the semilinear elliptic equation −∆u = f (u) in the whole R 2m , where f is of bistable type. It is known that in dimension 2m = 2 there exists a saddle-shaped solution. This is a solution which changes sign in R 2 and vanishes only on {|x 1 | = |x 2 |}. It is also known that this solution is unstable.In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension 2m = 4. More precisely, our main result establishes that if 2m = 4, every solution vanishing on the Simons cone {(x 1 , x 2 ) ∈ R m × R m : |x 1 | = |x 2 |} is unstable outside of every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.must be hyperplanes, at least if n ≤ 8. The conjecture has been proven to be true when the dimension n = 2 by Ghoussoub and Gui [16], and when n = 3 by Ambrosio and Cabré [5]. For 4 ≤ n ≤ 8 and assuming an additional limiting condition on u, it has been established by Savin [21] (see section 2 for more details).

Nondegeneracy of the saddle solution of the Allen-Cahn equation

“…In particular, in all even dimensions, saddle type solutions were found in [4]. These solutions vanish along the Simons cone.…”

“…Under the assumption that f (u) = 2u − 2u 3 , a computer assisted proof of this fact is given in [14]. In this respect, we would like to point out that the saddle-type solutions vanishing along Simons' cone in dimensions n = 4, 6 were found to have infinite Morse index ( [4], [3]). On the other hand, nondegeneracy of the saddle solution should imply that its Morse index is finite for any of the nonlinearity satisfying (1.2).…”

Saddle solutions of nonlinear elliptic equations involving the p-Laplacian

“…Recently, Cabré and Terra proved the existence of saddle solutions of bistable diffusion equations −Δu = f (u) in all even dimensions. They also obtained the instability of saddle solutions in R 4 and R 6 , see [6,7]. Before introducing the notion of saddle solution, we give the weak solution notion.…”

“…A natural candidate is expected to be founded in the class of saddle solutions. For this aim, Cabré and Terra [6] studied the existence of saddle solutions of −Δu = f (u) in every even dimension and instability of its saddle solutions in low even dimensions. Extending the Laplace operator to p-Laplace operator, we will consider the existence of saddle solutions of (1.1) in this paper.…”

Solutions to the Allen Cahn Equation and Minimal Surfaces

“…Savin [33] Monneau [27] and by Cabré and Terra [10]. See also the survey article by Farina and Valdinoci [22].…”