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MiS Preprint
64/1998
On a conjecture of Wolansky
Guofang Wang and Jun-Cheng Wei
Abstract
In this paper, we consider the following problem $(P_M)$: $$\delta u +\lambda |x|^\beta e^u =0, \ \ \ e \in B_R ,$$ $$-\int_{\partial B_R } \frac {\partial u }{\partial v } = M,$$ $$u=0 \ on \ \partial B_R$$ where $\lambda$ is an unknown constant, $\beta > 0$, $B_R = \{x \in R^2||x|<R\}$, $M$ is a prescribed constant and $v$ is the outer normal to the disk. Problem $(P_M)$ arises in the evolution of self-interacting clusters and also in prescribing Gaussian curvature problem. It is known that for $M < 8 \pi $, problem $(P_M)$ has a global minimizer solution (which is radially symmetric). We prove that for $M>8\pi$, there exists a $\beta_c \geq 1$ such that for $\beta > \beta_c $ and $M\in (8\pi ,4(2+\beta)\pi)\backslash\{m\pi,m=2,...\}$, problem $(P_M)$ 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.
