Liouville systems I: Blow-up analysis and orthogonal decomposition

Abstract

We study blow-up solutions of generalized Liouville systems on a closed surface at and near critical parameter hypersurfaces. Under a Morse assumption on the reduced functional, we establish the following results. First, we introduce amplitude variables in the approximate manifold and prove that every sequence of blow-up solutions at a fixed critical parameter, or along an admissible sequence from the lower side, admits a unique minimizing decomposition up to permutation. The remainder converges to zero and is orthogonal to the center, scale, and amplitude directions. The concentration scales are comparable, the limiting configuration satisfies the scale balance conditions, and the amplitudes vanish after multiplication by the logarithmic scale. Second, we determine the leading parameter defect in both critical decay regimes and prove that, along each fixed admissible path, all sufficiently large solutions lie in finitely many disjoint bubbling neighborhoods. Finally, assuming the leading coefficient is nonzero at every balanced critical configuration, the solution set at the fixed critical parameter is compact. These results provide the analytic foundation for studying the resonant degree via critical points at infinity.