Title: On a planar Hartree-Fock type system involving the (2,q)-Laplacian in the zero mass case

Abstract: In this talk, we investigate a class of planar Hartree-Fock type systems involving the (2,q)-Laplacian operator. Due to the nature of the problem, we deal with the logarithmic integral kernel. We introduce a suitable function space to study the system variationally. It is established existence of positive vector ground state solutions. In addition, we study the asymptotic behavior of the solutions, and we also provide a nonexistence result.

Title: Liouville theorem for a quasilinear non-uniformly elliptic equation in half-spaces and applications

Abstract: In this lecture, we will show Liouville-type theorems for a class of quasilinear equations in half-spaces, focusing on the classification of C^1-viscosity solutions that vanish on the flat boundary and grow linearly at infinity. This study connects to a broader class of divergence-form equations in Orlicz spaces, analyzed without the usual structural conditions, offering new insights and techniques for handling problems in generalized Sobolev spaces.

The proof strategy relies on three main components: regularity theory for solutions, the construction of barriers along the boundary, and novel gradient estimates up to the boundary. These tools enable a method that links Lipschitz regularity to higher-order smoothness. Additionally, we will discuss relevant applications, including the recovery of classical results such as Radó's zero-level set removability theorem and the Schwarz reflection principle, both now yielding substantially improved regularity results.

This is joint work with Diego Moreira (Federal University of Ceará) and Sergio Monari (Institute of Mathematical and Computer Sciences, University of São Paulo).

Title: Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Abstract: We establish the existence of nontrivial solutions for some class of quasilinear Schrödinger equations involving potentials that can be singular at the origin, unbounded, or vanishing at infinity, and the nonlinearity has critical exponential growth motivated by the Trudinger-Moser inequality. To prove our main result, we apply variational methods together with careful L∞-estimates.

Title: Choquard type equations with asymptotically linear nonlinearities

Abstract: In this talk,, we are going to present a class nonlocal elliptic problems in RN (N=>3) involving a positive continuous potential and the nonlinearity f(x,u) is asymptotically linear at infinity and at the origin in a suitable sense. The nonlocal term Iα ∗ F is the convolution of the Riesz potential with the primitive F(x, u) of the nonlinearity f(x, u). Our main results rely on the fact that nonlocal semilinear elliptic problems have nontrivial solutions whenever a kind of crossing of eigenvalues is allowed. We will consider an eigenvalue elliptic problem with a nonlocal term driven by the Choquard equation.

Title: Uma condição suciente para limitar sequências de Cerami em funcionais de Euler-Lagrange de problemas elípticos superlineares em domínios limitados

Abstract: Faremos umas discussões históricas apresentando hipóteses clássicas para limitar uma sequência de Cerami de funcionais de Euler-Lagrange associados à problemas de Dirichlet não-lineares em domínios limitados. Apresentamos uma condição mais geral que Ambrosetti-Rabinowitz.

Title: Some Hardy type inequalities and applications

Abstract: In this talk, we will present some recent results concerning Hardy-type inequalities and their applications. We will also deliver some existence results for a specific class of elliptic equations.

Title: The Generalized Fractional Sobolev Spaces: Advances and Theoretical Challenges

Abstract: In this talk, we will present recent results, theoretical advances, and challenges associated with generalized fractional Sobolev spaces. We will begin by discussing the foundations and motivations for studying these spaces, emphasizing their qualitative properties, embedding results, and applications to nonlocal problems. Finally, we will present some abstract results obtained, as well as the difficulties and limitations encountered in the development of this theory.

Title: Some critical elliptic equations involving perturbations

Abstract: In this talk, we study the effect of various perturbations on an elliptic problem with critical growth. One of the pioneering works in this line of research was developed by Brezis and Nirenberg in 1983, which has served as an inspiration for many authors.

[1] H. BREZIS AND L. NIRENBERG, solutions of nonlinear elliptic equations involving critical sobolev exponents, Communication on Pure and Applied Mathematics, 1983.