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).

We consider the elliptic equation −∆u = f (u) in the whole R 2m , where f is of bistable type. It is known that there exists a saddleshaped solution in R 2m . This is a solution which changes sign in R 2m and vanishes only on the Simons cone C = {(x 1 , x 2 ) ∈ R m ×R m : |x 1 | = |x 2 |}. It is also known that these solutions are unstable in dimensions 2 and 4.In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.which is monotone in one direction (say, for instance ∂ xn u > 0 in R n ), depends only on one Euclidean variable (equivalently, all its level sets are hyperplanes), at least if n ≤ 8. The conjecture has been proven to be true when the dimension n = 2 by Ghoussoub and Gui [22], and when n = 3

Abstract. In this paper we study the asymptotic behavior as time goes to infinity of the solution to a nonlocal diffusion equation with absorption modeled by a powerlike reaction −u p , p > 1 and set in R N . We consider a bounded, nonnegative initial datum u 0 that behaves like a negative power at infinity. That is, |x| α u 0 (x) → A > 0 as |x| → ∞ with 0 < α ≤ N . We prove that, in the supercritical case p > 1 + 2/α, the solution behaves asymptotically as that of the heat equation (with diffusivity a related to the nonlocal operator) with the same initial datum.

We study the large time behavior of nonnegative solutions of the Cauchy problem ut = J(x − y)(u(y, t) − u(x, t)) dy − u p , u(x, 0) = u0(x) ∈ L ∞ , where |x| α u0(x) → A > 0 as |x| → ∞. One of our main goals is the study of the critical case p = 1 + 2/α for 0 < α < N , left open in previous articles, for which we prove that t α/2 |u(x, t) − U (x, t)| → 0 where U is the solution of the heat equation with absorption with initial datum U (x, 0) = CA,N |x| −α . Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data u0 in the supercritical case and also in the critical case (p = 1 + 2/N ) for bounded and integrable u0.