Title: Multiplicity of Positive Solutions for Critical Kirchhoff Type Problems with Concave-Convex Nonlinearity in RN

Abstract: In this talk, we establish the existence and multiplicity of positive solutions for a critical elliptic problem in the whole space ℝⁿ. The main feature of this work is the treatment of a Kirchhoff-type elliptic problem where the nonlinearity is critical. Our primary goal is to investigate the existence and multiplicity of positive solutions for a Kirchhoff-type elliptic problem with a critical nonlinearity given by

−m(‖∇u‖₂²)u + V(x)u = λ f(x)|u|^{q−2}u + θ|u|^{2*−2}u, x ∈ ℝⁿ, u ∈ H¹(ℝⁿ). (Pc)

Here, the parameters satisfy λ, θ > 0, N ≥ 3, and q < 2 < 2(σ + 1) < 2*. The potential V: ℝⁿ → ℝ is bounded from below by a positive constant and belongs to L_loc(ℝⁿ).

This is joint work with Olímpio H. Miyagaki (UFSCar), José C. O. Junior (UFTN), and Eduardo Lima (UFSCar).

Title: Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms

Abstract: In this talk we present some results on the existence and multiplicity of solutions for a critical growth anisotropic quasilinear elliptic system, depending on two real parameters λ and μ. The system is coupled through a subcritical perturbation term and involves a negative critical part. We identify a specific scaling for the system and introduce a parameter γ associated with this scaling, which governs the geometry of the corresponding variational functional. We establish three distinct types of multiplicity results, depending on the regimes γ = 1, γ > 1, and γ < 1. The multiplicity results are based on a recent abstract critical point theorem for symmetric functionals on product spaces when γ ≥ 1, and on a truncation argument when γ < 1.

Title: On a critical Schrodinger equation involving singular and vanishing potentials in RN

Abstract: In this lecture, we present results concerning the existence and nonexistence of solutions to the following class of elliptic equations:

−Δu + |x|ᵃu = |x|ᵇ|u|^{2*(b)−2}u + λ|x|ᶜ|u|^{p−2}u, in ℝⁿ. (P)

Here N ≥ 3, 2*(b) := 2(N + b)/(N − 2), with a, b, c ≥ −2, and λ ∈ ℝ is a real parameter. The quantities 2*(b) and 2*(c) are critical exponents associated with this class of problems.

Using a Pohozaev-type identity, we establish a nonexistence result for problem (P) in the case p = 2*(c). Furthermore, when p < 2*(c) and under appropriate conditions on the parameters a and c, we prove the existence of weak solutions to (P). The proof relies on variational methods, particularly minimization techniques that require sharp estimates of the minimization level, as well as symmetry arguments based on the principle of symmetric criticality.

This research is joint work with Everaldo Souto de Medeiros (UFPB) and Uberlandio Batista Severo (UFPB), and it is currently under review for publication.

Title: Maximizers for the Logarithmic Trudinger-Moser functional beyond the critical threshold

Abstract: Our main goal is to show the existence of critical points for the functional associated with the logarithmic weighted Trudinger-Moser inequality, as introduced by M. Calanchi and B. Ruf in (J. Differential Equations, 258 (2015) 1967–1989), particularly beyond the critical constant. Our argument relies on the Lions concentration-compactness principle and some ideas presented by M. Struwe in (Ann. Inst. Henri Poincaré C, 5, (1988) 425-464).

Title: Advances in solving nonlinear Schrödinger equations with general potentials

Abstract: We present recent results on the existence of positive solutions to the nonlinear Schrödinger equation

−Δu + V(x)u = f(u), in ℝⁿ. (PV)

The results hold for very general potentials V, which may have positive or zero limits at infinity, allowing convergence from above, from below, or oscillatory behavior, without imposing any decay rate assumptions. The nonlinearities f may satisfy mild conditions, including superlinear or asymptotically linear growth at infinity.

This work is a collaboration with Romildo Lima (UFCG, Brazil) and Liliane Maia (UnB, Brazil). 

Title: On solutions for strongly coupled critical elliptic systems on compact Riemannian manifolds

Abstract: In this talk, we study the existence of solutions to a class of strongly coupled elliptic systems with critical growth on compact Riemannian manifolds. More precisely, by means of variational methods, we investigate the solvability of the system

−Δ_g u + a(x)u + b(x)v = (α / 2*) f(x) u |u|^{α−2} |v|^{β}, in M,
−Δ_g v + b(x)u + c(x)v = (β / 2*) f(x) v |v|^{β−2} |u|^{α}, in M.

Here (M, g) is a smooth closed Riemannian manifold of dimension n ≥ 3, Δ_g denotes the Laplace–Beltrami operator, a, b, and c are Hölder continuous functions on M, and f is a smooth function. The real parameters α > 1 and β > 1 satisfy α + β = 2*, where 2* = 2n / (n − 2) is the critical Sobolev exponent.

The analysis highlights how strong coupling effects and critical nonlinearities interact with the geometry of the manifold, leading to new existence results for elliptic systems on compact Riemannian manifolds.

This work is a collaboration with Nadiel de Oliveira Sousa (UFPI).

Title: Perturbations that transform the critical Brezis-Nirenberg problem into a supercritical one

Abstract: We analyze perturbations of the classical Brezis–Nirenberg problem that turn the equation into a supercritical one. While the original critical formulation admits no solutions in certain cases, the perturbed version generates positive radial solutions. In dimensions greater than or equal to three, this mechanism produces solutions in parameter ranges where nonexistence was expected. This is a joint work with Luiz Faria (UFJF-Brasil) and Jeferson Silva (UFOP- Brasil).

Title: Existence results for some elliptic problems in RN including Variable exponents above the critical growth

Abstract: In this work, we present existence results for a class of elliptic equations involving variable exponents:

−Δu + u = |u(x)|^{p(|x|)−1}u(x) + λ|u(x)|^{q(|x|)−1}u(x), x ∈ ℝⁿ.

Here λ ≥ 0, N ≥ 3, and p, q : [0, +∞) → (1, +∞) are radial continuous functions satisfying suitable conditions. For our purposes, it is sufficient to assume either subcriticality or criticality in a small region near the origin. Surprisingly, outside this region the nonlinearity may oscillate between subcritical, critical, and supercritical growth in the Sobolev sense.

Our approach allows the use of variational methods to address problems with variable exponents in ℝⁿ without imposing restrictions outside a neighborhood of the origin.

This work is a collaboration with Eudes Barboza (UFRPE), José Carlos de Albuquerque (UFPE), and Pedro Ubilla (USACH).