Radial Symmetry and Nondegeneracy for Singular Liouville Systems with Small Singular Exponents

We extend the classification (Lin and Zhang, Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 2010) to finite-mass singular Liouville systems with sufficiently small positive singular exponents. For symmetric, entrywise nonnegative, irreducible and invertible coupling matrices, every solution whose mass vector lies near a fixed compact subset of the regular mass surface is radial about the singular point and is uniquely determined by its component masses up to dilation. We also establish nondegeneracy modulo dilation. To overcome the failure of the moving-plane argument for positive exponents, we use compactness and a local reduction based on integral identities. The results also hold for nonzero nonnegative exponent vectors and for small mixed-sign vectors satisfying an explicit integral inequality.