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

Takahashi, Futoshi

Adv. Differ. Equ. 7, No. 3, 365-384 (2002).

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)

Keywords:

Aleksandrov-Bakelman-Pucci-type maximum principle; comparison principle between \(L^p\)-subsolutions and \(L^p\)-strong supersolutions; viscosity solutions; Hölder regularity