Preprint 64/1998

On a conjecture of Wolansky

Guofang Wang and Jun-Cheng Wei

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Submission date: 05. Oct. 1999 (revised version: October 1999)
Pages: 13
published in: Nonlinear analysis / A, 48 (2002) 7, Ser. A: Theory Methods, p. 927-937 
Keywords and phrases: semilinear equation, exponential nonlinearity, free energy functional, total curvature, conical singularity
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In this paper, we consider the following problem tex2html_wrap_inline10


where tex2html_wrap_inline18 is an unknown constant, tex2html_wrap_inline20, tex2html_wrap_inline22, M is a prescribed constant and tex2html_wrap_inline26 is the outer normal to the disk. Problem tex2html_wrap_inline28 arises in the evolution of self-interacting clusters and also in prescribing Gaussian curvature problem. It is known that for tex2html_wrap_inline30, problem tex2html_wrap_inline28 has a global minimizer solution (which is radially symmetric). We prove that for tex2html_wrap_inline34, there exists a tex2html_wrap_inline36 such that for tex2html_wrap_inline38 and tex2html_wrap_inline40, problem tex2html_wrap_inline28 admits a non-radially symmetric solution. This partially answers a conjecture of Wolansky. Our main idea is a combination of Struwe's technique and blow-up analysis for a problem with degenerate potential.

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