A double well potential system

Abstract

A semilinear elliptic system of PDEs with a nonlinear term of double well potential type is studied in a cylindrical domain. The existence of solutions heteroclinic to the bottom of the wells as minima of the associated functional is established. Further applications are given, including the existence of multitransition solutions as local minima of the functional.

Existence of a Heteroclinic Solution for a~Double Well Potential Equation in an Infinite Cylinder of ℝ N

“…Here we would like point out that the same arguments found in [8,Proposition 2.14] work to show that β > 0. Having this in mind, we can assume without loss of generality that…”

“…From this, U is a weak solution of (PDE). A regularity argument from [8,Section 6] implies that U ∈ C 2 (Ω, R), and that U is a classical solution of From this, U is a heteroclinic solution from 1 to -1, which finishes the proof of Theorem 1.1.…”

“…thereby showing that (Ũ n ) is also a minimizing sequence for J on Γ with In the next section, our main goal is proving that U is the desired heteroclinic solution, and in this point, the conditions on function A play their role. However, before doing that we need to say that if A is 1-periodic in x, the same arguments explored in [8] guarantee the existence of a heteroclinic solution W * from 1 to −1.…”

“…As in the above papers, we have used variational method, more precisely minimization technical on a special set, however new ideas have been introduced in the study of the problem, see for example, Proposition 3.1 in Section 2. The regularity and behavior of the heteroclinic are obtained by using the same arguments found in [8].…”

On the heteroclinic connection problem for multi-well gradient systems

“…that do not satisfy assumption (h). On the other hand, in the recent work [14] existence is obtained for a C 1 potential satisfying our assumptions (A1)-(A3) without (A4).…”