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Analysis/PDE’S seminar

Coordenador: Abiel Costa Macedo

Our Analysis/PDE’S seminar starts at 8.30 am and can be accessed by clicking on the date.

06/08/2020 ABSTRACT 

In this talk, we shed new light on the classical inequality by G.H. Hardy 1920 and discuss some related extremal problems. We aim to connect it with existence results for a wide range of differential equations on radially symmetric domains, including k-Hessian operator for fully nonlinear regime k > 1, and either critical or supercritical growth nonlinearities for both exponential and pure power-type growths.

Title: Hardy-type inequality and k-Hessian equations
Speaker: José Francisco de Oliveira(DMAT/UFPI)
13/08/2020 ABSTRACT 

In the talk we consider a class of elliptic problems with a nonlinearity which is nonlocal and with homogeneous Dirichlet boundary condition. Moreover the nonlinearity can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map, to produce multiplicity of classical positive solutions with ordered norms.

Title: Positive solutions for a singular problem
Speaker: Gaetano Siciliano(IME/USP)  
20/08/2020 ABSTRACT 

In the talk we deal with Hamiltonian elliptic systems in two dimensions and bounded domains, with one of the nonlinearities having exponential growth condition. We derive the maximal growth conditions allowed for the other one, proving that it can be of exponential type, double-exponential type, or completely arbitrary, depending on the conditions required for the first one. Under these hypotheses, we prove existence of nontrivial solutions for the system.

Title: Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions
Speaker: Bruno Henrique Carvalho Ribeiro(DMAT/UFPB)
27/08/2020 ABSTRACT 

In this talk, I'm going to present a variational approach for the M.S.P system using the concentration-compactness principal presented by P.l.lions in his famous paper on this subject.

Title: Existence of Steady States For The Maxwell-Schrodinger-Poisson System by The Concentration-Compactness Principal.
Speaker: Gabriel Neves Cunha(IME/UFG)
03/09/2020 ABSTRACT 

In this talk, we deal with an autonomous non-Lipschitz semilinear elliptic equation. We study the existence of a weak solution that satisfies some specific boundary conditions in some subset of the boundary, while in the rest of the boundary the Hopf maximum principle is violated, the so-called "free boundary solutions". In approaching this problem, the main techniques to the proof are the Pohozaev's type of identity and the generalized nonlinear Rayleigh quotients method.

Title: On free boundary periodic solutions for equations with non-Lipschitz nonlinearity
Speaker: Fábio Sodré Rocha(IME/UFG)


In this seminar, we will first show that the interaction between the Kirchhoff operator and the critical term leads to some variational properties of the energy functional as the sequential weak lower semicontinuity and the Palais–Smale condition provided a and b satisfy a suitable constraint.

Then, through a careful analysis of the fiber maps associated to the energy functional, we will prove existence, non-existence and multiplicity of solutions of our problem when the parameters a, b, λ vary in appropriate intervals. When the nonlinearity g is a pure power term, i.e. g(x, u) = |u|p−2u for some p ∈ (2, 2*), through a detailed study of the Nehari sets associated to the problem, we will show the existence of two critical hyperbolas on the plane (a, b) that separates the plane into regions where the energy functional exhibits distinct topological properties.

Speaker: Francesca Faraci(Department of Mathematics and Computer Sciences/ UNICT)


In this work, we study the existence of positive solutions for a class of fractional Schrödinger equations involving a potential function, which can vanish at infinity and super critical exponents. Our approach is based on a truncation argument in combination with variational techniques and the Moser iteration method.

Title: Fractional Schrödinger Equations with Potential Vanishing at Infinity and Supercritical Exponents
Speaker: Uberlândio Batista Severo(DMAT/UFPB)

24/09/2020 ABSTRACT 

In this talk, we review some well-known results for the eigenvalues of the Dirichlet p-Laplace operator. After, we study the existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional g-Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks
must be overcome, leading to some results that contrast with the case of power functions. The analysis developed in this talk extends the abstract framework corresponding to some standard cases associated with the p-Laplacian and the fractional Laplacian. We also address some perspectives and open questions.

Title: Variational Eigenvalues: Comparison between local and nonlocal operator
Speaker: Sabri Bahrouni(UTM)  


08/10/2020 ABSTRACT 

In this talk we shall consider a quasilinear elliptic problem involving the nonlocal p-Laplacian operator. The main feature here is to exhibit a positive solution using the well-known Nehari method, see [1]. At the end, we shall comment how these nonlocal elliptic problems have gained strength attention in the last few years, see also [2, 3]. Here also present some connections with the associated fibering maps, see for instance [4]. 


[1] Lou, Q. Luo, H. Multiplicity and concentration of positive solutions for fractional p−Laplacian problem involving concaveconvex nonlinearity, Nonlinear analysis: Real world applications, 387-405, 2018. 
[2] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des sciences mathmatiques, 521-573, 2012. 
[3] V. Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential, Eletronic journal of differential equations, 1-12, 2016. 
[4] E. D da Silva, M. L. M. Carvalho, C. Goulart, Critical quasilinear elliptic problems using concave-convex nonlinearities, Annali di matematica pura ed applicata, 693-726, 2018. 

Title: Positive solutions for a fractional p-Laplacian problem involving concave-convex nonlinearity
Speaker: Jefferson Luis Arruda Oliveira (IME/UFG) 


It is established the existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem de ned in the whole space ℜN. The work is devoted to studying a class of potentials and nonlinearities that can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by αΔ2u+βΔu+V(x)u where α≥0, β∈ℜ and V: ℜN→ℜ is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases negative, zero or positive. In order do that we employ some fine estimates proving the compactness for the associated energy functional.

Title: Asymptotically periodic fourth-order Schrodinger  equations with critical and subcritical growth

Speaker: Claudiney Goulart(UFJ)