Speaker: Kaye Silva
Universidade Federal de Goiás, Brazil

Date: December 16, 2021 at 16:15 Santiago time

Abstract:  Given an one-parameter family of C1-functionals, Φμ : X →R, defined on an uniformly convex Banach space X, we describe a method that permit us find critical points of Φμ at some energy level c ∈ R. In fact, we show the existence of a sequence μ(n,c), n ∈N, such that Φμ(n,c) has a critical level at c ∈ R, for all n ∈ N. Moreover, we show some good properties of the curves μ(n,c), with respect to c (for example, they are Lipschitz), and as a consequence of this analysis, we recover many know results on the literature concerning bifurcations of elliptic partial differential equations. Furthermore we prove new results for a large class of elliptic partial differential equations, which includes, for example, Ouyang, Lane-Enden, Concave-Convex, Kirchhoff and Schrödinger-Bopp-Podolsky type equations.

Venue: Online via Zoom
Chair: Gabrielle Nornberg

Speaker: Felipe Linares
IMPA, Brazil

Date: December 9, 2021 at 16:15 Santiago time

Abstract: In this talk we will discuss recent results regarding local well-posedness for the generalized Zakharov–Kuznetsov equation. We prove a high-dimensional version of the Strichartz estimates for the unitary group associated with the free Zakharov-Kuznetsov equation. As a by-product, we deduce maximal estimates which allow us to prove local well-posedness for the generalized Zakharov-Kuznetsov equation in the whole subcritical case whenever d\ge 4, k\ge 4 complementing the recent results of Kinoshita and Herr-Kinoshita. Finally, we use some of those maximal estimates in order to prove pointwise convergence results for the flow of the generalized Zakharov–Kuznetsov equation in any dimension, in the same spirit of a recent manuscript by Compaan, Lucà and Staffilani.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz

Speaker: Delia Schiera

Instituto Superior Técnico, Lisbon

Date: November 11, 2021 at 16:15 Santiago time

Abstract: I will present some existence results for a class of Choquard equations in which the potential has a positive limit at infinity and satisfies suitable decay assumptions. Also, it is taken invariant under the action of a closed subgroup of linear isometries of RN. As a consequence, the positive solution found is invariant under the same action. I will mainly focus on the physical case involving a quadratic nonlinearity.

Joint work together with Liliane Maia and Benedetta Pellacci.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Gabrielle Nornberg

Speaker: Jacopo Schino

Institute of Mathematics of the Polish Academy of Sciences, Poland

Date: October 28, 2021 at 16:15 Santiago time

Abstract: I will present a multiplicity result for the mixed dispersion non-linear Schrödinger equation

\[\Delta^2u−\beta \Delta u=g(u), \qquad \mbox{in}\quad\mathbb{R}^N \]

focusing on the case N \geq 5, where the non-linearity $g\colon\mathbb R\to \mathbb R$ satisfies assumptions in the spirit of Berestycki & Lions.After showing some compactness results, I will demonstrate how the variational approach of [1], which makes use of auxiliary functionals, can be used for this problem.

Venue: Online via Zoom
Chair: Michal Kowalczyk

Speaker: Rayssa Cajú

Universidade Federal da Paraíba, Brazil

Date: October 7, 2021 at 16:15 Santiago time

Abstract: The connections between geometry and partial differential equations have been extensively studied in the last decades. In particular, some problems arising in conformal geometry, such as the classical Yamabe problem, can be reduced to the study of PDEs with critical exponent on manifolds. More recently, the so-called Q-curvature equation, a fourth-order elliptic PDE with critical exponent, is another class of conformal equations that has drawn considerable attention by its relation with a natural concept of curvature. In this talk, I would like to discuss how fixed point methods can be helpful to study the Q-curvature equation in a singular setting, and discuss some interesting problems related to this topic.

Joint work with J.H. Andrade, J. M do O, J. Ratzkin and A. Silva Santos.

Venue: Online via Zoom
Chair: Gabrielle Nornberg