Speakers
Title: On the radial symmetry of positive solutions to doubly critical \(p\)-Laplacian systems
Abstract:
This talk focuses on a family of Hardy-Sobolev doubly critical \(p\)-Laplace systems defined on the entire Euclidean space, with the main goal of proving the radial symmetry of positive weak solutions.
A key tool in our analysis is the moving plane method, which serves as our starting point. We will review the state of the art for this technique in the context of more general problems than the seminal one treated by Gidas, Ni, and Nirenberg, culminating in the system under consideration.
Particular attention will be paid to how the quasilinear nature of the operator, the presence of the Hardy potential, and the coupled structure of the system combine, making the application of the method highly nontrivial.
The talk is based on joint work with Francesco Esposito, Rafael Lopez-Soriano and Berardino Sciunzi.
Title: Infinitely many sign-changing solutions for a nonlinear Neumann problem with critical exponents
Abstract:
We consider a doubly critical nonlinear Neumann problem on the half-space, which generalizes the equation for prescribing constant scalar and boundary mean curvature.
Positive solutions have been completely classified and are essentially unique, up to translations and dilations. On the other hand, we show the existence of infinitely many sign-changing solutions.
Compared to similar problems in the entire space, variational methods do not seem to work because of the smaller symmetry group of the half-space. We construct solutions using a Lyapunov-Schmidt reduction.
This is based on joint work in progress with G. Vaira (Bari).
Title: Existence and Obstructions in the \(Q\)- and \(T\)-Curvature Prescription Problem on \(\mathbb S^4_+\)
Abstract:
We consider the problem of prescribing nonconstant interior \(Q\)-curvature and boundary \(T\)-curvature on the four-dimensional upper hemisphere by means of a conformal change of the standard metric.
This problem leads to a fourth-order nonlinear elliptic equation in the interior, coupled with a third-order nonlinear boundary condition and a homogeneous Neumann condition.
We first investigate obstructions to solvability. Using conformal variations generated by boundary-preserving conformal vector fields, we derive Kazdan-Warner-type identities involving both the interior and boundary curvatures.
These identities provide necessary conditions for the existence of solutions and yield explicit nonexistence results.
We then address the existence of solutions through a mean-field-type variational formulation. Although the associated functional is bounded from below, it is generally noncoercive because of the noncompact action of the conformal transformation group of \(\mathbb S^4_+\).
By imposing suitable symmetry assumptions, we recover compactness and obtain a solution as a global minimizer when the prescribed curvatures are nonnegative.
As part of this analysis, we establish global and local higher-order Moser-Trudinger inequalities adapted to the structure of the problem.
Title: A bunch of bifurcation in the liquid drop model
Abstract:
In this talk, we consider the liquid drop model for atomic nuclei, in which the energy is given by the sum of surface tension and nonlocal repulsion under a volume constraint.
While balls are natural critical points, we show that, for an unbounded sequence of radii, non-spherical solutions bifurcate from the family of balls.
These solutions can have arbitrarily large volume.
Title: A fractional Loewner-Nirenberg result
Abstract:
The aim of the talk will be to extend the Loewner-Nirenberg type results for the Yamabe equation to the fractional case.
That is, given a prescribed set of certain codimension, we will analyze whether we are able to construct solutions which explode in this set.
An interesting issue that we will discuss is precisely what we understand by singular solution in the nonlocal setting.
