Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential

Abstract

We consider a class of semilinear elliptic system of the form: $$\begin{aligned} -\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in {\mathbb {R}}^{2}, \end{aligned}$$ (0.1) where \(W:{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) is a double well potential with minima \(\mathbf{a}_\pm \in {\mathbb {R}}^2\). We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system \(-\ddot{q}(x)+\nabla W(q(x))=0,\ x\in {\mathbb {R}}\), up to translations, is finite and constituted by not degenerate functions, then Eq. (0.1) has infinitely many solutions \(u\in C^{2}({\mathbb {R}}^{2})^{2}\), parametrized by an energy value, which are periodic in the variable y and satisfy \(\lim _{x\rightarrow \pm \infty }u(x,y)=\mathbf{a}_{\pm }\) for any \(y\in {\mathbb {R}}\).