Asymmetric heteroclinic double layers

MAPLY, CNRS et Université Claude Bernard, 69622 Villeurbanne Cedex, France; schatz@numerix.univ-lyon1.fr.

Received: 14 January 2002

Abstract

Let W be a non-negative function of class C³ from to , which vanishes exactly at two points a and b. Let S¹(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy is finite. Assume that there exist two isolated minimizers z₊ and z_- of the energy E₁ over S¹(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z₊ and z_-, it is possible to prove that there exists a function u from to itself which satisfies the equation and the boundary conditions The above convergences are exponentially fast; the numbers m₊ and m_- are unknowns of the problem.