Advances in Differential Equations

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

Futoshi Takahashi

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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

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

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


Takahashi, Futoshi. Multiple solutions of {$H$}-systems on some multiply-connected domains. Adv. Differential Equations 7 (2002), no. 3, 365--384.

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