Solutions with multiple catenoidal ends to the Allen–Cahn equation in R3

Abstract

We consider the Allen–Cahnequation Δu+u(1−u2)=0inR3. We construct two classes of axially symmetric solutions u=u(|x |,x3)suchthat the (multiple) components of the zero set look for large |x |like catenoids, namely|x3|∼Alog|x |.In Theorem 1 ,we find a solution which is even in x3, with Morse index one and a zero set with exactly two components,which are graphs.In Theorem 2,we construct a solution with a zero set with two or more nested catenoid-like components, whose Morse index become as large as we wish. While it is a common idea that nodal sets of the Allen–Cahn equation behave like minimal surfaces,these examples show that the non local interaction between disjoint portions of the nodal set,governed in suitably a symptotic regimes by explicit systems of 2dPDE, induces richness and complex structure of the set of entire solutions, beyond the one in minimal surface theory