

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
Bibtex
Keywords and phrases: semilinear equation, exponential nonlinearity, free energy functional, total curvature, conical singularity
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Abstract:
In this paper, we consider the following problem
where is an unknown constant,
,
, M is a prescribed constant and
is the outer normal to the disk. Problem
arises in the evolution of self-interacting clusters and also in prescribing Gaussian curvature problem. It is known that for
, problem
has a global minimizer solution (which is radially symmetric). We prove that for
, there exists a
such that for
and
, problem
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.