Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation

Authors:

Institution:

(1) Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo S. Leonardo Murialdo, 1-00146 Rome, Italy;(2) Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

Abstract:

For the Neumann sinh-Gordon equation on the unit ball ${B \subset \mathbb {R}^2}$

$\left\{ \begin{array}{ll} -\Delta u = \lambda^+ \left( \frac{e^u}{\int_B e^u}-\frac{1}{\pi} \right)-\lambda^- \left( \frac{e^{-u}}{\int_B e^{-u}}-\frac{1}{\pi} \right) & {\rm in}\,B\\ \frac{\partial u}{\partial \nu}=0 & {\rm on}\, \partial B \end{array} \right.$

we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ^± are positive parameters. It answers partially an open problem formulated in Jost et al. Calc Var Partial Diff Equ 31(2):263–276]. The research of the first named author is supported by M. U. R. S. T., project “Variational methods and nonlinear differential equations”. The research of the second named author is supported by an Earmarked grant from RGC of Hong Kong.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35J60 35B33 35J25 35J20 35B40

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