Classification of finite Morse index solutions to the elliptic sine-Gordon equation in the plane

Abstract

The elliptic sine-Gordon equation is a semilinear elliptic equation with a special double well potential. It has a family of explicit multiple-end solutions. We show that all finite Morse index solutions belong to this family. It will also be proved that these solutions are nondegenerate, in the sense that the corresponding linearized operators have no nontrivial bounded kernel. Finally, we prove that the Morse index of $2n$-end solutions is equal to $n(n-1)/2$.