We consider the SU(3) singular Toda system on a compact surface (Σ,g)−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑m=1Mα1m(δpm−1)−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑m=1Mα2m(δpm−1), where hi are smooth positive functions on Σ, ρi∈R+, pm∈Σ and αim>−1. We give both existence and non-existence results under some conditions on the parameters ρi and αim. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohožaev-type identities."

VL - 270 UR - http://www.sciencedirect.com/science/article/pii/S0022123615004942 ER - TY - JOUR T1 - Moser–Trudinger inequalities for singular Liouville systems JF - Mathematische Zeitschrift Y1 - 2016 A1 - Luca Battaglia AB -In this paper we prove a Moser–Trudinger inequality for the Euler–Lagrange functional of general singular Liouville systems on a compact surface. We characterize the values of the parameters which yield coercivity for the functional, hence the existence of energy-minimizing solutions for the system, and we give necessary conditions for boundedness from below. We also provide a sharp inequality under assuming the coefficients of the system to be non-positive outside the diagonal. The proofs use a concentration-compactness alternative, Pohožaev-type identities and blow-up analysis.

VL - 282 UR - https://doi.org/10.1007/s00209-015-1584-7 ER - TY - JOUR T1 - Existence and multiplicity result for the singular Toda system JF - Journal of Mathematical Analysis and Applications Y1 - 2015 A1 - Luca Battaglia KW - Existence result KW - Liouville-type equations KW - Multiplicity result KW - PDEs on compact surfaces KW - Toda system AB -We consider the Toda system on a compact surface (Σ,g)−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑j=1Jα1j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑j=1Jα2j(δpj−1), where hi are smooth positive functions, ρi are positive real parameters, pj are given points on Σ and αij are numbers greater than −1. We give existence and multiplicity results, using variational and Morse-theoretical methods. It is the first existence result when some of the αij's are allowed to be negative."

VL - 424 UR - http://www.sciencedirect.com/science/article/pii/S0022247X14010191 ER - TY - JOUR T1 - A general existence result for the Toda system on compact surfaces JF - Advances in Mathematics Y1 - 2015 A1 - Luca Battaglia A1 - Aleks Jevnikar A1 - Andrea Malchiodi A1 - David Ruiz KW - Geometric PDEs KW - Min–max schemes KW - Variational methods AB -In this paper we consider the following Toda system of equations on a compact surface:−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−Δu1=−4π∑j=1mα1,j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−Δu2=−4π∑j=1mα2,j(δpj−1), which is motivated by the study of models in non-abelian Chern–Simons theory. Here h1,h2 are smooth positive functions, ρ1,ρ2 two positive parameters, pi points of the surface and α1,i,α2,j non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation−Δu=ρ1(heu∫ΣheudVg−1)−ρ2(he−u∫Σhe−udVg−1), which arises in fluid dynamics."

VL - 285 UR - http://www.sciencedirect.com/science/article/pii/S0001870815003072 ER - TY - JOUR T1 - A note on compactness properties of the singular Toda system JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Y1 - 2015 A1 - Luca Battaglia A1 - Gabriele Mancini AB -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.

VL - 26 U1 - 34669 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - THES T1 - Variational aspects of singular Liouville systems Y1 - 2015 A1 - Luca Battaglia KW - Variational methods, Liouville systems, Moser-Trudinger inequalities, min-max methods AB - I studied singular Liouville systems on compact surfaces from a variational point of view. I gave sufficient and necessary conditions for the existence of globally minimizing solutions, then I found min-max solutions for some particular systems. Finally, I also gave some non-existence results. PB - SISSA U1 - 34737 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - A Moser-Trudinger inequality for the singular Toda system JF - Bull. Inst. Math. Acad. Sin. Y1 - 2014 A1 - Luca Battaglia A1 - Andrea Malchiodi VL - 9 ER - TY - JOUR T1 - Remarks on the Moser–Trudinger inequality JF - Advances in Nonlinear Analysis Y1 - 2013 A1 - Gabriele Mancini A1 - Luca Battaglia AB -We 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.

PB - Advances in Nonlinear Analysis VL - 2 UR - http://edoc.unibas.ch/43974/ IS - 4 N1 - The article is composed of 32 pages inad recorded in PDF format U1 - 34666 U2 - Mathematics U4 - 1 U5 - MAT/05 ER -