Existence and multiplicity result for the singular Toda system

We consider the Toda system on a compact surface. We give existence and multiplicity results, using variational and Morse-theoretical methods. It is the first existence result when some of the coefficients of the singularities are allowed to be negative.Comment: 37 pages, 1 figure, accepted on Journal of Mathematical Analysis and Application

Moser-Trudinger inequalities for singular Liouville systems

In this paper we prove a Moser-Trudinger inequality for the Euler-Lagrange functional of a general singular Liouville system. We characterize the values of the parameters which yield coercivity for the functional and we give necessary conditions for boundedness from below. We also provide a sharp inequality under some assumptions on the coefficients of the system.Comment: 22 pages, Accepted on Mathematische Zeitschrif

Uniform bounds for solutions to elliptic problems on simply connected planar domains

We consider the singular Liouville equation and the Henon-Lane-Emden problem on simply connected planar domains. We show that any solution to each problem must satisfy a uniform bound on the mass. The same results applies to some systems and more general non-linearities. The proofs are based on the Riemann mapping theorem and a Pohozaev-type identity.Comment: 9 pages, accepted on Proceedings of the American Mathematical Societ

A general existence result for stationary solutions to the Keller-Segel system

We consider a Liouville-type PDE on a smooth bounded planar domain, which is related to stationary solutions of the Keller-Segel's model for chemotaxis. We prove existence of solutions under some algebraic conditions on the parameters. In particular, if the domain is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.Comment: 21 pages, accepted on Discrete and Continuous Dynamical System

A Moser-Trudinger inequality for the singular Toda system

In this paper we prove a sharp version of the Moser-Trudinger inequality for the Euler-Lagrange functional of a singular Toda system, motivated by the study of models in Chern-Simons theory. Our result extends those for the scalar case, as well as for the regular Toda system. We expect this inequality to be a basic tool to attack variationally the existence problem under general assumptions.Comment: 13 pages, accepted on Bulletin of the Institute of Mathematica Academia Sinic

Remarks on the Moser-Trudinger inequality

We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincar\'e'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.Comment: 32 pages, accepted on Advances in Nonlinear Analysi

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

We consider the SU(3) Toda system on a compact surface. We give both existence and non-existence results under some conditions on the parameters. 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 Pohozaev identities.Comment: 41 pages, 9 figures, accepted on Journal of Functional Analysi

Existence of groundstates for a class of nonlinear Choquard equations in the plane

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$-\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2,$$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the plane $\mathbb{R}^2$ under general nontriviality, growth and subcriticality on the nonlinearity $F \in C^1 (\mathbb{R},\mathbb{R})$.Comment: revised version, 16 page

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

We consider a nonlinear Choquard equation $$-\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N,$$ when the self-interaction potential $V$ is unbounded from below. Under some assumptions on $V$ and on $p$, covering $p =2$ and $V$ being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $u\in H^1 (\mathbb{R}^N)\setminus\{0\}$ by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.Comment: 16 page