Luca Battaglia

Existence and multiplicity result for the singular Toda system

Abstract We consider the Toda system on a compact surface ( Σ , g ) { − Δ u 1 = 2 ρ 1 ( h 1 e u 1 ∫ Σ h 1 e u 1 d V g − 1 ) − ρ 2 ( h 2 e u 2 ∫ Σ h 2 e u 2 d V g − 1 ) − 4 π ∑ j = 1 J α 1 j ( δ p j − 1 ) , − Δ u 2 = 2 ρ 2 ( h 2 e u 2 ∫ Σ h 2 e u 2 d V g − 1 ) − ρ 1 ( h 1 e u 1 ∫ Σ h 1 e u 1 d V g − 1 ) − 4 π ∑ j = 1 J α 2 j ( δ p j − 1 ) , where h i are smooth positive functions, ρ i are positive real parameters, p j are given points on Σ and α i j 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 α i j s are allowed to be negative.

A note on compactness properties of the singular Toda system

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

Remarks on the Moser–Trudinger inequality

Abstract. We extend the Moser–Trudinger inequality to any Euclidean domain satisfying Poincarés 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 .

Existence of Groundstates for a Class of Nonlinear Choquard Equations in the Plane

Abstract We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equations - Δ ⁢ u + u = ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u )   in ⁢ ℝ 2 , -\Delta u+u=\bigl{(}I_{\alpha}*F(u)\bigr{)}F^{\prime}(u)\quad\text{in }\mathbb% {R}^{2}, where I α {I_{\alpha}} is the Riesz potential of order α on the plane ℝ 2 {\mathbb{R}^{2}} under general nontriviality, growth and subcriticality on the nonlinearity F ∈ C 1 ⁢ ( ℝ , ℝ ) {F\in C^{1}(\mathbb{R},\mathbb{R})} .

Groundstates of the Choquard equations with a sign-changing self-interaction potential

We consider a nonlinear Choquard equation

A MOSER-TRUDINGER INEQUALITY FOR THE SINGULAR TODA SYSTEM

\begin{aligned} -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text {in }\mathbb {R}^N, \end{aligned}

Moser-Trudinger inequalities for singular Liouville systems

-Δu+u=(V∗|u|p)|u|p-2uinRN,when the self-interaction potential V is unbounded from below. Under some assumptions on