Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential (Q523355)

The authors study the semilinear elliptic system \(-\Delta u+\nabla W(u)=0\) on the plane \(\mathbb{R}^2\), where \(W: \mathbb{R}^2\to\mathbb{R}\) is a double-well potential with minima \(a_+\) and \(a_-\). It is proved that if the set of minimal heteroclinic solutions to the one-dimensional system \(\ddot q(x)+ \nabla W(q(x))= 0\) on \(\mathbb{R}\), up to translations, is finite and formed by nondegenerate functions, then there are infinitely many solutions in \(C^2(\mathbb R^2)^2\) which are parametrized by an energy value, which are periodic in the \(y\) variable, and such that \(\lim_{x\to\pm\infty} u(x,y)= a_+, a_-\) for any \(y\) in \(\mathbb{R}\). The proof is given by using variational methods, where problems due to the lack of compactness must be overcomed.