MiS Preprint Repository

In this paper, we consider the following problem $(P_M)$: $$\delta u+\lambda |x{|}^{\beta }{e}^u=0,e\in B_R,$$ $$-{\int }_{\partial B_R}\frac{\partial u}{\partial v}=M,$$ $$u=0on\partial B_R$$ 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 ${\beta }_c\ge 1$ such that for $\beta >{\beta }_c$ and $M\in (8\pi ,4(2+\beta )\pi )\mathrm{\setminus }\{m\pi ,m=2,...\}$, 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.