1,692 research outputs found
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 where is the Riesz potential of order on
the plane under general nontriviality, growth and subcriticality
on the nonlinearity .Comment: revised version, 16 page
Groundstates of the Choquard equations with a sign-changing self-interaction potential
We consider a nonlinear Choquard equation when
the self-interaction potential is unbounded from below. Under some
assumptions on and on , covering and being the one- or
two-dimensional Newton kernel, we prove the existence of a nontrivial
groundstate solution 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
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