A note on compactness properties of the singular Toda system

Battaglia, Luca; Mancini, Gabriele

doi:10.4171/rlm/708

Cited by 14 publications

(31 citation statements)

References 16 publications

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“…More precisely, if we have blow up at a regular point x R ∈ M \ S for a sequence (u n ) n of solutions to (1.2), it holds where a k will be defined in (1.4). By some further analysis, see for example [6,4,8,34], from the above quantization result it follows that the set of solutions to (1.2) is uniformly bounded in C 2,β for any fixed β ∈ (0, 1) provided that ρ / ∈ Σ. Thus, the Leray-Schauder degree d ρ of (1.2) is well-defined for ρ / ∈ Σ.…”

Section: Introductionmentioning

confidence: 90%

On the topological degree of the mean field equation with two parameters

Jevnikar¹,

Wei²,

Yang³

2018

Indiana Univ. Math. J.

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We consider the following class of equations with exponential nonlinearities on a compact surface M :which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here h 1 , h 2 are smooth positive functions and ρ 1 , ρ 2 are two positive parameters.We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided ρ i / ∈ 8πN, i = 1, 2. Then we study the blow up behavior when ρ 1 crosses 8π and ρ 2 / ∈ 8πN. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the SU (3) Toda system, we can compute the Leray-Schauder topological degree for ρ 1 ∈ (0, 8π) ∪ (8π, 16π) and ρ 2 / ∈ 8πN. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.2000 Mathematics Subject Classification. 35J20, 35J60, 35R01. Key words and phrases. Geometric PDEs, Mean field equation, Topological degree. A.J. is partially supported by the PRIN project Variational and perturbative aspects of nonlinear differential problems.

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Section: Introductionmentioning

confidence: 90%

On the topological degree of the mean field equation with two parameters

Jevnikar¹,

Wei²,

Yang³

2018

Indiana Univ. Math. J.

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“…Using Green's representation formulas, in [7] it was proved that in case of blow-up at least one component u i must accumulate at a finite number of points, and therefore the corresponding limiting parameter ρ i must be quantized, according to Theorem 1.4. As a consequence, one finds the following result.…”

Section: Theorem 14 ([25])mentioning

confidence: 99%

Min–max schemes for SU(3) Toda systems

Malchiodi

2016

J. Fixed Point Theory Appl.

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This paper surveys some recent results on Toda systems of Liouville equations. These systems model self-dual non-abelian ChernSimons vortices, and arise in the study of holomorphic curves. Suitable min-max schemes are employed, leading to existence of solutions in several situations. We will use in particular properties about concentration of exponential functions in order to describe low-energy levels of the Euler-Lagrange energy.Mathematics Subject Classification. 35B33, 35J35, 53A30, 53C21.

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“…We need to show that f ≡ 0, which will be done arguing as in [8] (Lemmas 2.2 and 2.3). If f ≡ 0, then one easily sees that f = V e w , with V := ρ…”

Section: Corollary 22mentioning

confidence: 99%

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

Battaglia¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis:where Ω ⊂ R 2 is a smooth bounded domain and β, ρ are real parameters. We prove existence of solutions under some algebraic conditions involving β, ρ. In particular, if Ω is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.(1.1)Since both (P β,ρ ) and (1.1) are invariant under addition of constants, it will not be restrictive to look for solutions to (P β,ρ ) satisfyingˆΩ u = 0; equivalently, we will consider J β,ρ not on its naturalIn particular, we will study the topology and the homology of very low energy sub-levelswith c = −L ≪ 0, then will deduce existence of solutions using Morse theory. This argument has been introduced in [17] for a fourth-order problem in Riemannian geometry and it has become a rather classical tool in the study of Liouville-type equation (see for instance the surveys [26,25]). With respect to most previous results, the main new difficulties are given by the Neumann boundary conditions, rather than Dirichlet, and the presence of the extra linear term βu.Neumann conditions may cause concentration on ∂Ω, which was excluded in the case of Dirichlet conditions, whereas concentration on the interior is similar in the two cases. The main difference when concentration occurs at the boundary is due to the fact that, roughly speaking, a shrinking ball centered at a point p ∈ ∂Ω is asymptotically half of a shrinking ball contained in Ω. The argument to fix such an issue is inspired by [28], which deals with a similar problem on higherdimensional manifolds with boundary.Another issue may be given by linear part −∆ + β not being positive definite, if β < 0, which is a new feature in second order Liouville equations. This naturally leads to consider the projection of u into positive and negative sub-spaces of the operator −∆ + β. Precisely, we take an orthonormal frame {ϕ i } i∈N of eigenfunctions of −∆ with associated positive non-decreasing eigenvalues {λ i } i∈N (counted with multiplicity), so thatNow, if −β is not an eigenvalue of −∆, then −λ I+1 < β < −λ I for some I ∈ N, therefore we can define the projection Π I on a finite dimensional subspace, on which orthogonal −∆ + β is positive definite:Arguing as in [17], we can show that, if J β,ρ (u) ≪ 0, then either e ú Ω e u is concentrated around a finite number of points or Π I u is large (or both occur). To express this alternative we will use the join, which has been used in the variational study of Liouville system of two equations, where one has an alternative between the concentration of each component (see [6,4,20,7,5]). Given two topological spaces X and Y , its join X × Y is defined as the product between the two spaces and the unit interval, with identifications at each endpoint. We setwith Σ being a compact surface with boundary and h ∈ C ∞ (Σ) being strictly positive.Here, if h is not constant, we can also cover the case ...

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A note on compactness properties of the singular Toda system

Cited by 14 publications

References 16 publications

On the topological degree of the mean field equation with two parameters

On the topological degree of the mean field equation with two parameters

Min–max schemes for SU(3) Toda systems

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

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