In this note, we consider blow-up for solutions of the SU(3) Toda system on compact surfaces. In particular, we give a complete proof of a compactness result stated by Jost, Lin and Wang and we extend it to the case of singular systems. This is a necessary tool to find solutions through variational methods.

1 aBattaglia, Luca1 aMancini, Gabriele uhttps://www.math.sissa.it/publication/note-compactness-properties-singular-toda-system00769nas a2200109 4500008004100000245006200041210006100103260001600164520036100180100002200541856009600563 2015 en d00aOnofri-Type Inequalities for Singular Liouville Equations0 aOnofriType Inequalities for Singular Liouville Equations bSpringer US3 aWe study the blow-up behavior of minimizing sequences for the singular Moser–Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence of at most two singularities.

1 aMancini, Gabriele uhttps://www.math.sissa.it/publication/onofri-type-inequalities-singular-liouville-equations01019nas a2200121 4500008004100000245008500041210006900126260001000195520053500205653002000740100002200760856011500782 2015 en d00aSharp Inequalities and Blow-up Analysis for Singular Moser-Trudinger Embeddings.0 aSharp Inequalities and Blowup Analysis for Singular MoserTruding bSISSA3 aWe investigate existence of solutions for a singular Liouville equation on S^2 and prove sharp Onofri-type inequalities for a Moser-Trudinger functional in the presence of singular potentials. As a consequence we obtain existence of extremal functions for the Moser-Trudinger embedding on compact surfaces with conical singularities. Finally we study the blow-up behavior for sequences of solutions Liouville-type systems and prove a compactness condition which plays an important role in the variational analysis of Toda systems.10aMoser-Trudinger1 aMancini, Gabriele uhttps://www.math.sissa.it/publication/sharp-inequalities-and-blow-analysis-singular-moser-trudinger-embeddings00685nas a2200097 4500008004100000245008200041210006900123520032200192100002200514856005100536 2015 en d00aSingular Liouville Equations on S^2: Sharp Inequalities and Existence Results0 aSingular Liouville Equations on S2 Sharp Inequalities and Existe3 aWe prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on S^2 in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.

1 aMancini, Gabriele uhttp://urania.sissa.it/xmlui/handle/1963/3448900793nas a2200145 4500008004100000245004800041210004800089260003500137300001200172490000600184520038200190100002200572700002000594856003300614 2013 en d00aRemarks on the Moser–Trudinger inequality0 aRemarks on the Moser–Trudinger inequality bAdvances in Nonlinear Analysis a389-4250 v23 aWe extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincaré's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.

1 aMancini, Gabriele1 aBattaglia, Luca uhttp://edoc.unibas.ch/43974/