On a conjecture of Wolansky
Guofang Wang and Jun-Cheng Wei
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Submission date: 05. Oct. 1999 (revised version: October 1999)
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
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.