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Classification of radial solutions to Liouville systems with singularities

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  • Let $A=(a_{ij})_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: \begin{eqnarray*} \left\{ \begin{array}{lcl} \Delta u_i+\sum_{j=1}^n a_{ij}|x|^{\beta_j}e^{u_j(x)}=0,\quad \mathbb R^2, \quad i=1,...,n\\ \\ \int_{\mathbb R^2}|x|^{\beta_i}e^{u_i(x)}dx<\infty, \quad i=1,...,n \end{array}\right. \end{eqnarray*} where $\beta_1,...,\beta_n$ are constants greater than $-2$. If all $\beta_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate.
    Mathematics Subject Classification: Primary: 35J47; Secondary: 35J60.


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