Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

Abstract

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

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.…”

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

“…We will present it for the solution u ε of the stretched problem (51). It follows from (129) that, if ε is small, the function ϕ * = u ε − u ap satisfies…”

“…We emphasize that this is a key step in our construction of an approximate solution for (51). Our underlying plan, carried out in the following subsection, is to smoothly interpolate betweenũ out and u in , and u in and zero (in R 2 \D ε ), near the curveΓ ε , in order to obtain an approximate solution u ap of (51) that is valid in all of R 2 .…”

“…We can now define the inner solution of problem (51), in the neighborhood of Γ ε described by {|x| δ 0 ε…”

“…In this subsection we will construct a smooth approximate solution for problem (51) that is valid in all of R 2 . This will be achieved by smoothly interpolating betweenũ out and u in inD ε , and between u in and zero in R 2 \D ε .…”