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Open Access Research

Multiple blowing-up and concentrating solutions for Liouville-type equations with singular sources under mixed boundary conditions

Yibin Chang* and Haitao Yang

Author Affiliations

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

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Boundary Value Problems 2012, 2012:33  doi:10.1186/1687-2770-2012-33

Published: 23 March 2012

Abstract

In this article, we mainly construct multiple blowing-up and concentrating solutions for a class of Liouville-type equations under mixed boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M1">View MathML</a>

for ε small, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M2">View MathML</a>, Ω is a bounded, smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3">View MathML</a>, Γ := {p1, ..., pN} ⊂ Ω is the set of singular sources, δp denotes the Dirac mass at p, ν denotes unit outward normal vector to Ω and b(x) > 0 is a smooth function on Ω.

2000 Mathematics Subject Classification: 35B25; 35J25; 35B38.

Keywords:
multiple blowing-up and concentrating solution; Liouville-type equation; singular source; mixed boundary conditions; finite dimensional reduction