## Advances in Differential Equations

- Adv. Differential Equations
- Volume 7, Number 3 (2002), 365-384.

### 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.

#### Article information

**Source**

Adv. Differential Equations Volume 7, Number 3 (2002), 365-384.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651830

**Mathematical Reviews number (MathSciNet)**

MR1867692

**Zentralblatt MATH identifier**

1126.35325

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35J20: Variational methods for second-order elliptic equations 35J50: Variational methods for elliptic systems 35J60: Nonlinear elliptic equations 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05]

#### Citation

Takahashi, Futoshi. Multiple solutions of {$H$}-systems on some multiply-connected domains. Adv. Differential Equations 7 (2002), no. 3, 365--384. https://projecteuclid.org/euclid.ade/1356651830