Elliptic PDEs

This thematic session focuses on recent advances in the theory of elliptic partial differential equations and their applications. We aim to bring together researchers working on qualitative and quantitative properties of solutions, variational methods, geometric aspects of elliptic problems, and related nonlinear phenomena. 

The session is aimed at graduate students, postdoctoral fellows, and researchers with an interest in elliptic PDEs and related areas. Our goal is to present recent developments, stimulate new collaborations, and offer an accessible overview of current trends in the field. 

Wednesday - January 21 - Morning

Speaker: Sandra Imaculada (Universidade Estadual do Maranhão)

Hour and room: 9:00-9:40 - Department of Mathematics - 2nd Floor - 347

Title: Um breve passeio pelas equações de Schrödinger quasilineares

Abstract: Faremos um breve apanhado histórico sobre as formas como esta  classe de equações  vem sendo estudada quando pesquisadores buscam sua solução, veja por exemplo  [1],[3],[4] e [5]. Essa classe de problemas tem sido bastante estudada nos últimos anos e generalizações dela vêm sendo propostas, nesta conversa temos o intuito também de exemplificar algumas delas, como [2] e [6].

[1] M. Colin and L. Jeanjean. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Analysis, 56 (2004), 213-226.

[2] J. C. Oliveira Junior and S. I. Moreira (2020): Generalized quasilinear equations with signchanging unbounded potential. Applicable Analysis, DOI:10.1080/00036811.2020.1836356

[3] J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations II. $J$. Differential Equations, 187 (2003), 473-493.

[4] J-q. Liu, Y-q. Wang and Z-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method. Communications in Partial Differential Equations, 29 (2004), 879-901.

[5] J-q. Liu, Y-q. Wang and Z-Q. Wang, Quasilinear elliptic equations via perturbation method. Proceedings of the American Mathematical Society, 141 (2012), 253-263.

[6] Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Analysis 80 (2013) 194-201.

Speaker: Eudes Mendes Barboza (Universidade Federal Rural de Pernambuco)

Hour and room: 9:45-10:25 - Department of Mathematics - 2nd Floor - 347

Title: Existence results for some elliptic problems in $\mathbb{R}^N$ including variable exponents above the critical growth

Abstract:  In this talk, we establish existence results for the following class of equations with variable exponents 

-\Delta u + u = |u(x)|^{p(|x|)-1}u(x) + \lambda |u(x)|^{q(|x|)-1}u(x), in \mathbb{R}^N,

where $\lambda \ge 0$, $N \ge 3$, and $p,q : [0,+\infty) \to (1,+\infty)$ are continuous radial functions satisfying  suitable assumptions. Our analysis shows that it is sufficient to impose either subcritical or critical growth only in a small neighborhood of the origin. Nevertheless, this framework allows the application of variational methods to problems with variable exponents in $\mathbb{R}^N$ without imposing any growth restrictions away from the origin. More precisely, outside this region, the nonlinearities are allowed to oscillate, in the Sobolev sense, between subcritical, critical, and supercritical regimes.

Speaker: Junior Bessa (Universidade Estadual de Campinas)

Hour and room: 10:30-11:10 - Department of Mathematics - 2nd Floor - 347

Title: Optimal Continuity Moduli for Fully Nonlinear Elliptic Models with Oblique Boundary Conditions

Abstract: In this talk, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by

F(D^2u,x) = f(x)    in  \Omega

\beta(x) \cdot Du(x) + \gamma(x) \, u(x) = g(x)     on   \partial \Omega.

Such regularity estimates are obtained by combining the integrability properties of $f$ in different regimes with a VMO-type assumption on the coefficients of $F$, together with suitable smoothness conditions on the boundary data $\beta, \gamma$ and $g$. In particular, we derive sharp estimates in the borderline cases $f \in L^n(\Omega)$ and $f \in p\text{-BMO}(\Omega)$. Moreover, for source terms in $L^p(\Omega)$, with $p \in (n,\infty)$, we establish sharp gradient estimates. Finally, we also obtain Schauder-type estimates formulated in the framework of Dini-type moduli of continuity. 

This is a joint work with João Vitor da Silva (Unicamp-Brazil) and Gleydson C. Ricarte (UFC- Brazil). 

Speaker: Diego Ferraz de Souza (Universidade Federal do Rio Grande do Norte)

Hour and room: 11:15-12:00 - Department of Mathematics - 2nd Floor - 347

Title: Semilinear elliptic problems involving a fast increasing diffusion weight

Abstract: In this talk, we discuss the existence and multiplicity of positive bounded solutions for a class of elliptic partial differential equations in the whole Euclidean space $\mathbb{R}^N$. The core of this work is the study of a divergence operator involving a fast increasing diffusion coefficient (e.g., with exponential growth). We address the scenario where the nonlinearity exhibits a mixed behavior, "concave" near the origin and "convex" for larger values. We show how to obtain a first solution using the sub- and supersolution method, adapting classical approximation techniques to handle the concave term. Furthermore, we present the existence of a second solution, larger than the first, via variational methods and the Mountain Pass geometry. A key highlight of our approach is a new compactness result established for the associated energy functional, which is crucial to overcome the lack of compactness typical of problems in unbounded domains involving fast increasing weights.

Wednesday - January 21 - Afternoon

Speaker: Lorena Maria Augusto Pequeno Silva (Universidade Federal de Pernambuco)

Hour and room: 14:00-14:40 - Department of Mathematics - 2nd Floor - 347

Title: On a weighted Adams-type inequality and an application to a polyharmonic equation

Abstract: In this talk, we have two objectives. The first deals with the improvement of a class of Adams-type inequalities involving potentials V and weights K, which can decay to zero at infinity as $(1 + |x|^\alpha)^{-1}, \alpha \in (0,N)$, and $(1 + |x|^\beta)^{-1}, \beta \in [\alpha,+ \infty) $ for all $x \in \mathbb{R}^N$, respectively. The second objective is to use minimax methods and the Adams inequality obtained in the first moment to establish the existence of solutions for the following class of problems:

\sum_{j=1} ^m (-\Delta)^j u + V (x)u = K(x)f(x, u)      in   \mathbb{R}^{2m},

where $(-\Delta) ^j$ denotes the polyharmonic operator, m is a positive integer, and the nonlinear term $f(x, u)$ can have critical exponential growth.

Speaker: Hugo Henryque Coelho e Silva (Universidade Federal de Pernambuco)

Hour and room: 14:45-15:10 - Department of Mathematics - 2nd Floor - 347

Title: Morse Theory and Lower Bound for the Morse Index of Nodal Radial Solutions for Gradient Elliptic Systems

Abstract: In this work, we present general results from Morse theory for functionals "I" defined on Hilbert spaces, recalling critical groups and reviewing some classification results. We emphasize the main motivations for studying the Morse index in relation to multiplicity and qualitative properties of critical points of "I". We then establish a lower bound for the Morse index of radial nodal solutions in a broad class of gradient-type elliptic systems. To this end, we connect the Morse index to the negative spectrum of the linearized operator and construct suitable nonradial eigenfunctions to produce negative directions for the associated quadratic form. Our main result shows that the Morse index exceeds its radial counterpart by a term depending on the space dimension and the number of nodal domains of each component. Finally, we discuss how this estimate reflects the structural properties of radial and nodal solutions.