Vol. 250, No. 1, 2011

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ISSN: 0030-8730
Mixed interior and boundary nodal bubbling solutions for a sinh-Poisson equation

Juncheng Wei, Long Wei and Feng Zhou

Vol. 250 (2011), No. 1, 225–256
Abstract

We consider here the semilinear equation Δu + 2ε2 sinhu = 0 posed on a bounded smooth domain Ω in R2 with homogeneous Neumann boundary condition, where ε > 0 is a small parameter. We show that for any given nonnegative integers k and l with k + l 1, there exists a family of solutions uε that develops 2k interior and 2l boundary singularities for ε sufficiently small, with the property that

               2∑k              ∑2l
2<i>ε</i>2 sinhu<i>ε</i> ⇀ 8π   (− 1)i−1δξi + 4π  (− 1)i−1δξi,
i=1              i=1

where (ξ1,2(k+l)) are critical points of some functional defined explicitly in terms of the associated Green function.

Keywords
boundary-interior nodal bubbling solutions, sinh-Poisson equation
Mathematical Subject Classification 2000
Primary: 35J20, 35J65
Milestones
Received: 3 January 2010
Accepted: 18 February 2010
Published: 1 March 2011
Authors
Juncheng Wei
Department of Mathematics
The Chinese University of Hong Kong
Room 220, Lady Shaw Building
Shatin, Hong Kong
Hong Kong
Long Wei
Institute of Applied Mathematics and Engineering Computations
Hangzhou Dianzi University
Hangzhou, Zhejiang 310018
China
Feng Zhou
Department of Mathematics
East China Normal University
Shanghai 200062
China