Multiple solutions of {$H$}-systems on some multiply-connected domains

Abstract

In this note, we consider the following problem: \begin{eqnarray*} \left \{ \begin{array}{l} {\Delta} u = 2 u_x \wedge u_y \quad \mbox{in} \; \Omega, \quad u \in {H_0^1(\Omega ; {\bf R}^3)}, \\ u |_{\partial \Omega} = 0, \end{array} \right. \end{eqnarray*} where $\Omega \subset {\bf R}^2$ is a smooth bounded domain. We show that if the domain $\Omega$ is conformal equivalent to a $(K+1)$-ply connected domain satisfying some conditions, then the problem has at least $K$ distinct non-trivial solutions.