Classification of radial solutions to  Liouville systems with singularities

Chang-Shou Lin; Lei Zhang

doi:10.3934/dcds.2014.34.2617

Article Contents

2014, Volume 34, Issue 6: 2617-2637. Doi: 10.3934/dcds.2014.34.2617

This issue Previous Article Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations Next Article Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$

Classification of radial solutions to Liouville systems with singularities

Chang-Shou Lin^1, and
Lei Zhang^2,

1.
Taida Institute of Mathematical Sciences and Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 106
2.
Department of Mathematics, University of Florida, 358 Little Hall, P.O.Box 118105, Gainesville, Florida 32611-8105

Received: August 31, 2012

Revised: May 31, 2013

Published: May 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J47; Secondary: 35J60.

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