## Multiple solutions of $$H$$-systems on some multiply-connected domains.(English)Zbl 1126.35325

Summary: In this note we consider the following problem $$-\Delta u=2u_x\wedge u_y$$ in $$\Omega$$, $$u=0$$ on $$\partial\Omega$$, where $$\Omega$$ is a bounded smooth domain in $$\mathbb R^2$$, $$u\in H_0^1(\Omega;\mathbb R^3)$$ and “$$\wedge$$” denotes the usual vector product in $$\mathbb R^3$$. 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 nontrivial solutions.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J50 Variational methods for elliptic systems 35J60 Nonlinear elliptic equations 58E12 Variational problems concerning minimal surfaces (problems in two independent variables)