Talks
Vieri Benci
Lucio Boccardo
Anna Maria Candela
Daniele Cassani
Edcarlos D. da Silva
João Henrique de Andrade
José Francisco de Oliveira
João Marcos do Ó
Carlos Alberto dos Santos
Ederson dos Santos
João Rodrigues dos Santos Júnior
Stefania Gatti
Short Talks
Vieri Benci (Università di Pisa) Title: Past and present of the Delta function Abstract: We propose a brief survey on the notion of generalized function with particular enphasis on the Delta function. Also we present some examples and open problems. |
Lucio Boccardo (Istituto Lombardo — Università di Roma Sapienza) Title: A Brezis-Merle inequality for solutions of nonlinear two-dimensional Dirichlet problems in \(L^1\) and applications Abstract: We study two-dimensional Dirichlet problems (linear and nonlinear) with discontinuous coefficients, order one terms and data in \(L^1\) (and no more). The focus is a Brezis-Merle inequality. |
Anna Maria Candela (Università di Bari) Title: Quasilinear elliptic equations on unbounded domains: a dichotomy result Abstract: Our aim is investigating the existence of bounded solutions of the quasilinear elliptic problem $$\textrm{(P)} \qquad \begin{cases} {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{in $\Omega$,}\\ u = 0 & \hbox{on $\partial\Omega$,} \end{cases}$$ with \(A_t(x,t,\xi) = \frac{\partial A}{\partial t}(x,t,\xi)\), \(a(x,t,\xi) = \nabla_\xi A(x,t,\xi)\) for a given \(A(x,t,\xi)\) which grows as \(|\xi|^p + |t|^p\), \(p > 1\), where \(\Omega \subseteq {\mathbb R}^N\), \(N \ge 2\), is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually \(\Omega = {\mathbb R}^N\). Under suitable assumptions on \(A(x,t,\xi)\) and \(g(x,t)\), problem \(\textrm{(P)}\) has a variational structure and admits at least one bounded solution which can be found by passing to the limit on a sequence \((u_k)_k\) of bounded positive solutions on bounded domains. Then, either such a solution is nontrivial or a constant \(\bar{\lambda} > 0\) and a sequence of points \((y_k)_k\) exist such that $$|y_k| \to +\infty\qquad \hbox{and}\qquad \int{B_1(y_k)} |u_k|^p dx \ge \bar{\lambda}\quad \hbox{for all $k \ge 1$} $$ with \(B_1(y_k) = {x \in {\mathbb R}^N:\ |x – y_k| < 1}\). Joint work with Giuliana Palmieri and Addolorata Salvatore. Partially supported by MUR–PRIN 2022 PNRR Project P2022YFAJH Linear and Nonlinear PDE’s: New directions and Applications” and INdAM – GNAMPA Project 2024 Nonlinear problems in local and nonlocal settings with applications”. |
Daniele Cassani (Università dell’Insubria) Title: Normalized solutions to NLS equations in dimension two Abstract: We are concerned with positive mass-normalized solutions to semi-linear Schrödinger equations. We focus on the so-called mass mixed case in which the nonlinearity has \(L^2\)-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: a local minimizer and a second one which is of mountain pass type. We also investigate the asymptotic behavior of solutions approaching the zero mass case, namely when the normalization constant vanishes. |
Abiel Costa Macedo (Universidade Federal de Goiás) Title: Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals Abstract: Let \(W^{m,\frac{n}{m}}(\mathbb{R}^n)\) with \(1\le m < n\) be the standard higher order derivative Sobolev space in the critical exponential growth threshold. We investigate a new Adams-Adimurthi-Druet type inequality on the whole space \(\mathbb{R}^n\) which is strongly influenced by the vanishing phenomenon. Specifically, we prove $$\sup_{\underset{|\nabla^{m} u|_{\frac{n}{m}}^{^{\frac{n}{m}}}+|u|_{\frac{n}{m}}^{\frac{n}{m}} \leq 1}{u\in W^{m,\frac{n}{m}}(\mathbb{R}^n)}} \int_{\mathbb{R}^n}\Phi\left(\beta \left(\frac{1+\alpha|u|_{\frac{n}{m}}^{\frac{n}{m}}}{1-\gamma\alpha|u|_{\frac{n}{m}}^{\frac{n}{m}}}\right)^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\right) \mathrm{d}x<+\infty. $$ where \(0\le \alpha<1\), \(0<\gamma<\frac{1}{\alpha}-1\) for \(\alpha>0\), \(\nabla^{m} u\) is the \(m\)-th order gradient for \(u\), \(0\le\beta\le \beta_0\), with \(\beta_0\) being the Adams critical constant, and \(\Phi(t) = e^{t}-\sum_{j=0}^{j_{m,n}-2}\frac{t^{j}}{j!}\) with \(j_{m,n}=\min{j\in\mathbb{N}\;:\: j\ge n/m}\). In addition, we prove that the constant \(\beta_0\) is sharp. In the subcritical case \(\beta<\beta_0\), the existence and non-existence of extremal function are investigated for \(n=2m\) and attainability is proven for \(n=4\) and \(m=2\) in the critical case \(\beta=\beta_0\). Our method to analyze the extremal problem is based on blow-up analysis, a truncation argument recently introduced by DelaTorre-Mancini and some ideas by Chen-Lu-Zhu, who studied the critical Adams inequality in \(\mathbb{R}^4\). |
Edcarlos D. da Silva (Universidade Federal de Goiás) Title: Positive solutions for a Kirchhoff type problem with critical growth via nonlinear Rayleigh quotient Abstract: In the present work we establish the existence and multiplicity of positive solutions for a critical elliptic problem in the whole space \(\mathbb{R}^N\). The main feature here is to treat a Kirchhoff-type elliptic problem where the nonlinearity is critical and defines a sign-changing function. Our approach relies on the minimization method applied to the Nehari method together with the nonlinear Rayleigh quotient method. Here we use the fibering map associated with the energy functional which exhibits degenerate points under suitable values on the two parameters within the nonlinearity. This difficulty does not allow us to apply the Lagrange Multipliers Theorem in general. |
João Henrique de Andrade (Universidade de São Paulo) Title: Quantitative stability of the total \(Q\)-curvature near minimizing metrics Abstract: We discuss new quantitative stability estimates for the total \(Q\)-curvature functional of order \(k\in \mathbb{N}\) near minimizing metrics on any smooth, closed (compact and without boundary) \(n\)-dimensional Riemannian manifold, where \(k \in (1, \frac{n}{2}) \cap \mathbb{Z}\) and \(n\in \mathbb{N}\). More precisely, under suitable positivity assumptions, we show that (generically) the distance to the set of minimizing metrics is quadratically controlled by the energy deficit of the \(Q\)-curvature, extending recent results for the scalar curvature case \(k = 1\). In the degenerate setting, we further prove the existence of Riemannian manifolds where this quadratic control can be improved: to cubic when \(k \in {2, 3}\), and to quartic for general \(k \in (1, \frac{n}{2}) \cap \mathbb{Z}\). These degenerate examples appear to be of independent interest and provide a framework for constructing solutions to the total \(Q\)-curvature flows that converge at slow (polynomial) rates. |
Ranieri de França Freire (Université de Pau et des Pays de l’Adour) Title: Weighted Sobolev Trace Embeddings and Their Applications Abstract: This study presents new results on weighted Sobolev trace embeddings and investigates their applications to a class of quasilinear elliptic equations with nonlinear boundary conditions. Existence results are established via the fibering method, while Liouville-type results are obtained through a priori estimates. These nonlinear elliptic equations are motivated by challenges that arise in diverse scientific domains, notably in physics and applied mathematics. |
José Francisco de Oliveira (Universidade Federal do Piauí) Title: Concentration level of the Adams functional and maximizers Abstract: In this talk, we will present a sharp estimate for the level of the Adams functional acting along normalized concentrating sequences. Our estimate extends the pioneering one by Carleson and Chang to the higher-order derivative setting. Furthermore, we will apply this estimate to investigate the extremal problem for the Adams inequality under the Navier boundary conditions due to C. Tarsi. |
João Marcos do Ó (Universidade Federal da Paraíba) Title: Supercritical Trudinger–Moser Inequalities and Applications Abstract: The Trudinger–Moser inequality, as the critical limiting case of Sobolev embeddings, is fundamental in the study of nonlinear PDEs. We present new Trudinger–Moser-type inequalities on the unit disk demonstrating a two-phase asymptotic structure: near the origin, the functional follows the classical Moser profile with non-compactness manifested through blow-up phenomena; near the boundary, it transitions to a Bliss-type regime characterized by concentration shocks. These results yield new insights into elliptic problems with critical and supercritical nonlinearities. |
Carlos Alberto dos Santos (Universidade de Brasília) Title: Connected sets of solutions to problems with singular regions Abstract: In this talk, we will present an abstract result that is used to show the existence of a connected set of solutions for problems that do not necessarily have a priori boundedness of solutions and may present regions of singularities. We apply this new result to establish the existence of connected branches of strongly positive classical solutions for Dirichlet problems headed by quasilinear Schrödinger- and Carrier-type operators. A fine qualitative study of these connected branches is also presented. |
Ederson dos Santos (Universidade de São Paulo) Title: Standing waves for nonlinear Hartree type equations — Radial symmetry and classification Abstract: In this talk I will report some results on standing wave solutions for nonlinear Hartree type equations, regarding the radial symmetry for positive solutions and a classification result for (positive) ground state solutions. Joint work with with Eduardo Böer and Gustavo Ramos (ICMC-USP). |
João Rodrigues dos Santos Júnior (Universidade Federal do Pará) Title: Nonexistence results and multiple solutions for \(p\)-Laplacian nonlocal problem Abstract: We establish nonexistence results for a nonlocal \(p\)-Laplacian problem, generalizing recent work in the field. Additionally, we prove a multiplicity result that provides conditions for obtaining a prescribed number of positive solutions through suitable geometric constraints on a one-dimensional fixed-point map. Computable examples illustrating our main results are included. |
Gustavo Ferron Madeira (Universidade Federal de São Carlos) Title: Elliptic equations with Sobolev supercritical and upper Hardy-Littlewood-Sobolev critical exponents Abstract: We study the existence of nontrivial nonnegative solutions for parametric elliptic equations that include a potentially supercritical (in the Sobolev sense) term and a nonlocal term with the Hardy-Littlewood-Sobolev critical exponent. Under certain conditions, we determine the exact parameter range where nonnegative solutions exist (or do not exist). In a slightly more restricted interval a second nontrivial solution is constructed. Moreover, when the nonlinear term has concave behavior near the origin, infinitely many solutions with the asymptotic behavior of its energy are obtained. The results, inspired by the seminal work of Alama and Tarantello (1996) on local problems in bounded domains, combine variational methods, Leray-Schauder degree theory, and Krasnoselskii’s genus via biorthogonal functionals in separable and reflexive Banach spaces. Joint work with Olímpio H. Miyagaki (UFSCar, Brazil) and Patrizia Pucci (Università degli Studi di Perugia, Italy). Partial support from FAPESP, Projeto Temático TESEd: Temático em Equações e Sistemas de Equações diferenciais – 2022/16407-1. |
Stefania Gatti (Università di Modena e Reggio Emilia) Title: Recent results on the 2D Cahn-Hilliard equation with nondegenerate mobility Abstract: In this talk I will discuss some recent results on the 2D Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential obtained in a joint work with Monica Conti, Pietro Galimberti and Andrea Giorgini. In particular, we proved that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. |
Jacques Giacomoni (Université de Pau et des Pays de l’Adour) Title: Existence and nonexistence results for a weighted elliptic problem in half space Abstract: We consider a weighted ellitic problem posed in the half space of the form: $$ \begin{cases} -\mathrm{div}(\rho(x_N) \nabla u) &=a|u|^{p-2}u &\mbox{in }& \mathbb{R}^N_+, \\ -\frac{\partial u}{\partial x_N}&=b|u|^{q-2}u&\mbox{on }& \partial \mathbb{R}^N_+ =\mathbb{R}^{N-1}. \end{cases} $$ We study regularity, existence and nonexistence of weak solutions for this problem. Some results contrast to former results about the case where $\rho=1$ It is a joint work with Joao Marcos Do Ó, Ranieri F. Freire, Everaldo S. Medeiros. |
Massimo Grossi (Università di Roma Sapienza) Title: Boundary regularity results for degenerate elliptic operators Abstract: In this talk, we will discuss some regularity results for solutions to the degenerate Poisson problem $$\begin{cases} -{\rm div}(d^\beta\nabla u)=f & {\rm in}\ \Omega \\ u=0 & {\rm on}\ \partial\Omega, \end{cases} $$ where \(d = \mathrm{dist}(x,\partial\Omega)\) is the distance from \(x\) to the boundary, and \(\beta < 1\). This operator becomes degenerate or singular near the boundary depending on the sign of \(\beta\), and this significantly influences the behavior of the solution. In a joint paper with Marta Calanchi (Università di Milano, Italy), we provide precise upper and lower estimates of the solution near the boundary, capturing its asymptotic behavior and determining the optimal Hölder regularity that can be expected. We also investigate whether the solution \(u\) belongs to standard Sobolev spaces, proving that \(\beta = \frac12\) is a threshold for the inclusion in \(W^{1,2}(\Omega)\). |
Masato Hashizume (Osaka University) Title: On compactness of Sobolev-type embeddings with variable exponents Abstract: In this talk, we study Sobolev-type embedding theorems in Sobolev spaces with variable exponents. We focus on embeddings into Lebesgue spaces with variable exponents, where the variable exponent takes the critical value at a single point and subcritical values elsewhere. Using a different approach from previous works, we prove that the compactness or non-compactness of the embedding is characterized by the asymptotic rate at which the exponent approaches the critical value. We also investigate Strauss-type embeddings in the whole space. We first provide a sufficient condition on the variable exponent to ensure continuous embedding, and then we establish a condition under which the embedding is compact. Similar problems can be considered for Trudinger-Moser type functionals, and we give a condition on the variable exponent that guarantees compactness. This talk is based on joint work with Michinori Ishiwata (The University of Osaka). |
Elard Juarez Hurtado (Universidade Estadual Paulista) Title: On Embeddings and Critical Phenomena in Anisotropic Spaces Abstract: We present an anisotropic extension of Lions’ concentration-compactness principle within the framework of anisotropic spaces. Our results establish fundamental properties, including compact embeddings, and provide a functional-analytic framework for the study of variational problems and critical elliptic equations in these anisotropic settings. This work unifies and extends previous embedding theorems, emphasizing the intrinsic role of anisotropy in the functional structure. |
Michinori Ishiwata (Osaka University) Title: On the soliton-resolution conjecture for semilinear parabolic problems in \(\mathbb{R}^N\) including energy critical case Abstract: In this talk, we consider the asymptotic behavior of solutions for semilinear parabolic problems with noncompact orbit. Let \(N\geq 3\), \(2^*:=\frac{2N}{N-2}\) be the critical Sobolev exponent and let $$ \textrm{(P)}\qquad \begin{cases} \partial_t u=\Delta u-\gamma u+u|u|^{p-2} & \mbox{ in }\mathbb{R}^N\times (0,\infty), \\ u|_{t=0}=u_0 & \mbox{ in }\mathbb{R}^N, \end{cases} $$ where \(u_0\in\dot{H}^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\) (for simplicity) and \(p\in (2,2^*]\). We put \(\gamma=1\) if \(p\in (2,2^*)\) and \(\gamma=0\) if \(p=2^*\). The existence of (time-global) solutions for \(\textrm{(P)}\) is now well-known. We are interested in the asymptotic behavior of solutions of \(\textrm{(P)}\) with noncompact orbit. The asymptotic behavior of solutions for such evolution equations with compact orbit is described by the so-called LaSalle principle which indicates that the omega-limit set is contained in the set of all stationary solutions, thus every global-in-time solution converges to a stationary solution along a time sequence. Note that the orbit of solutions of [/latex]\textrm{(P)}[/latex] may be noncompact in the natural energy space due to the invariance of the associated energy functional under the action of the translation for \(p\in(2,2^*]\) and the dilation for \(p=2^*\). In this case, the LaSalle principle may break down and one may expect that all the solutions of \(\textrm{(P)}\) could be a superposition of translated and dilated stationary solutions of \(\textrm{(P)}\). This conjecture is called the soliton-resolution conjecture. In this talk, we introduce several results that answer the soliton-resolution conjecture for both of the cases \(p<2^*\) and \(p=2^*\). The basic tool is the profile decomposition of the orbit along time sequence together with the variational structure. |
Guozhen Lu (University of Connecticut) Title: Sharp stability for geometric inequalities in the Euclidean spaces and the Heisenberg group Abstract: In this talk, we will discuss some recent developments on optimal stability for geometric and functional inequalities in both Euclidean spaces and the Heisenberg group. These include the asymptotically sharp stability for fractional Sobolev inequalties of all orders and the Hardy-Littlewood-Sobolev inequalities on the Euclidean spaces, and the optimal stability for the first order Sobolev inequality on the Heisenberg group. These are joint works with L. Chen and H. Tang. If time permits, we will also discuss joint works with Anh Do, C. Cazacu, J. Flynn, D. Ganguly, N. Lam and A. Rassunov on the stability for a class of Caffarelli-Kohn-Nirenberg inequalities and Gaussian Poincare inequalities in the Euclidean spaces (and hyperbolic spaces). |
Liliane Maia (Universidade de Brasília) Title: A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior Abstract: We consider the weakly coupled elliptic system of logistic type, $$ \begin{cases} -\Delta u &=\lambda_1 u- |u|^{p-2}u+ \beta |u|^{\frac{p}{2}-2}u |v|{^{\frac{p}{2}-1}}v & \mbox{ in }\Omega,\\ -\Delta v & =\lambda_2 v- |v|^{p-2}v+\beta |u|^{\frac{p}{2}-1}u|v|^{\frac{p}{2}-2}v & \mbox{ in }\Omega,\\ u,v &\in H_0^1(\Omega), \end{cases} \qquad \textrm{(LS)} $$ where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with \(N\geq 2\), \(2< p < 2^*\), and \(\lambda_1(\Omega)< \lambda_1 \leq \lambda_2\). We say the system is competitive if \(\beta<0\) and cooperative if \(\beta>0\), for \(\beta \in \mathbb{R}\). We prove the existence and multiplicity of solutions to the problem (LS) in alternative variational frameworks, depending on the range of the parameter \(\beta\). We do not rely on bifurcation or degree theory, which have been used in the literature for logistic-type problems. Instead, the novelty is to obtain min-max type solutions by exploiting the different geometry of the functional associated with the logistic problem. In case \(N\geq 2\) and suitable values of \(p\), we extend the existence results, for all \(\beta\) in the whole line, and possibly for the classical case \(N=3\) and \(p=4\). Furthermore, we analyze the asymptotic behavior of such solutions as \(\beta \to 0\) or \(\beta \to \pm \infty\). This is a work in collaboration with Haoyu Li (UFSCar, Brazil) and Mayra Soares (UnB, Brazil). |
Carlo Mercuri (Università di Modena e Reggio Emilia) Title: Solutions to nonlinear Schrödinger equations involving Coulombic interactions Abstract: I will discuss two separate classes of nonlinear Schrödinger equations (NLS) both involving Coulomb type interactions, and arising from the quantum many body problem. The first kind of results, dealing with the multiplicity of stationary solutions, have been recently obtained in collaboration with Kanishka Perera (California Institute of Technology) using scaling properties of functionals in an essential way. In the second part of the talk I will instead describe some uniqueness results obtained with collaborators from Eindhoven University of Technology on time dependent NLS equations coupled with classical nuclear dynamics. |
Anna Maria Micheletti (Università di Pisa) Title: Compactness and non compactness of the full set of solutions for Yamabe problem on manifolds with boundary Abstract: Let \((M,g)\) a compact Riemannian manifold of dimension \( n \) with umbilic boundary. It is well known that there exists, in the conformal class of the metric \( g \), at least one scalar-flat metric with a constant mean curvature on the boundary, and that when the manifold is of positive type, there are multiple solutions. We prove that the full set of these metrics is a compact set provided \(n>5 \) and the Weyl tensor is always different from zero on the boundary. We also give some result of stability (or non stability) of compactness under different types of perturbation of the problem. These results are in collaboration with Marco Ghimenti and Angela Pistoia. |
Enzo Mitidieri (Università di Trieste) Title: Liouville theorems and qualitative properties of semilinear elliptic inequalities in a half-space Abstract: We discuss several applications derived from the first part of the paper L. D’Ambrosio and E. Mitidieri, Characterization of positive superharmonic functions in a half-space, Preprint 2025, with a particular emphasis on the qualitative properties of solutions of integral and differential inequalities on a half-space, as well as the corresponding Liouville theorems. |
Olimpio H. Miyagaki (Universidade Federal de São Carlos) Title: Existence of two normalized solutions for a Choquard equation with exponential growth and an \(L^2\)-subcritical perturbation Abstract: This talk is concerned with the existence of normalized solutions for the following class of Choquard elliptic problems: $$ \begin{cases} -\Delta u + \lambda u = (I_{\alpha} \ast F(u))f(u) + \mu (I_{\alpha} \ast |u|^{q})|u|^{q-2}u, & \textrm{ in } \mathbb{R}^{2}, \\ \displaystyle\int_{\mathbb{R}^2}|u|^2 = a, \end{cases} $$ where \(a>0\), \(I_{\alpha}\) is the Riesz potential, \(\ast\) represents the convolution operator, \(f\) has exponential critical growth in \(\mathbb{R}^2\), \(F(t)= \int_{0}^t f(s)ds\), and \(1 + \frac{\alpha}{2} < q < 2 + \frac{\alpha}{2}\). By variational methods, we prove the existence of two normalized solutions, one with negative energy and one with positive energy. Talk related to the paper: Haoyu Li,Braulio Maia,O.H. Miyagaki- ZAMP 2024. |
Roberta Musina (Università di Udine) Title: Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights Abstract: We deal with weighted Hardy–Sobolev type inequalities for functions on \(\mathbb R^d\), \(d\ge 2\). The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions. This is a joint paper with Gabriele Cora (Université Libre de Bruxelles) and Alexander I. Nazarov (St. Petersburg Dep. of Steklov Inst. and St. Petersburg State University). |
Daisuke Naimen (Muroran Institute of Technology) Title: Infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs Abstract: We study infinite concentration and oscillation phenomena on supercritical elliptic equations in discs. We first detect an infinite sequence of bubbles on any blow-up solutions. The precise characterization of the profile, energy, and position of each concentrating part is given via the Liouville equation with the energy recurrence formulas. An immediate conclusion is that the infinite concentration breaks the uniform boundedness of the energy. Moreover, it leads to a description of the asymptotic shapes of the graphs of blow-up solutions which shows several infinite oscillation behaviors around singular solutions. This observation contains a proof of infinite oscillations of bifurcation diagrams which yield the existence of infinitely many solutions. |
Filomena Pacella (Università di Roma Sapienza) Title: Stability and asymptotic behavior of one-dimensional solutions of Lane-Emden problems Abstract: We study a local shape-optimization problem for the functional which represents the energy of positive solutions of a Lane-Emden relative Dirichlet problem in domains contained in an unbounded cylinder \(C\). Our aim is to understand how the energy of a nondegenerate positive solution behaves under small perturbations of the domain, inside \(C\). In particular we consider a bounded cylinder \(C(L)\) of height \(L>0\) and the corresponding positive one-dimensional solution \(u(L)\) which depends only on the variable which describes the height of the cylinder. Given the simple geometry of \(C(L)\) and of \(u(L)\), it would be reasonable to expect that \((C(L);u(L))\) would locally minimize the energy functional. We will show that this is not always the case, but the stability/instability properties of \((C(L);u(L))\) depend on the exponent \(p\) of the nonlinearity. We obtain quite precise results as \(p\) tends to \(1\) or to infinity. This is achieved by a careful asymptotic analysis of the one-dimensional solutions, which is of independent interest. The results are contained in the PhD Thesis of D.G.Afonso and in papers in collaborations with D.G.Afonso and A.Iacopetti and with F.De Marchis and L.Mazzuoli. |
Enea Parini (Aix-Marseille Université) Title: On Hopf’s Lemma for sign-changing supersolutions to fractional Laplacian equations Abstract: Unlike in the local case, Hopf’s Lemma does not hold true, in general, for sign-changing supersolutions to equations driven by the fractional Laplacian, as proven in a recent paper by Dipierro, Soave and Valdinoci. In this talk we will investigate the validity of Hopf’s Lemma for a (possibly sign-changing) function \(u \in H^s_0(\Omega)\) satisfying $$(-\Delta)^s u(x) \geq c(x)u(x) \quad \text{in }\Omega,$$ where \(\Omega \subset \mathbb{R}^N\) is an open, bounded domain, \(c \in L^\infty(\Omega)\), and \((-\Delta)^s u\) is the fractional Laplacian of \(u\). In particular, we will highlight the relation between the validity of Hopf’s Lemma for \(u\) at a point \(x_0 \in \partial \Omega\), and the validity of Hopf’s Lemma for the Caffarelli-Silvestre extension of \(u\) at the point \((x_0,0) \in \mathbb{R}^N \times \mathbb{R}^+\). The results have been obtained in collaboration with Azahara DelaTorre. |
Benedetta Pellacci (Università della Campania) Title: Optimizing the total population in some ”not-logistic” models Abstract: We study some optimization problems concerning the total population whose density solves an elliptic equation in the presence of a nonlinearity not of logistic type. This kind of models arises in the description of tumor cells growth. In this context, the sign-changing weight, usually describing the distribution of resources in the logistic model, corresponds to the treatment of the tumor. We address the issue of maximizing or minimizing the total population size with respect to the resources distribution, considering some point-wise, not uniform, bounds as well as prescribing the total amount of treatment. We prove that, for some special class of growths, the minimal configuration is constant and any maximal configuration is bang-bang (extreme point of the admissible set). As a consequence, this problem can be recast as a shape optimization problem, the unknown domain standing for the treatment location. In the one-dimensional case, we deepen the analysis, showing some qualitative properties of the unknown optimal set. This is a joint collaboration with Iulia Martina Bulai (Università di Torino) and Francesca Gladiali (Università di Sassari). |
Paolo Piccione (Universidade de São Paulo) Title: Conformal curvatures and Geometric PDEs Abstract: In this talk I will discuss how notions of curvature evolve under conformal changes of metric, with a focus on their interplay with nonlinear elliptic partial differential equations. After reviewing classical and Riemannian curvature, I will turn to conformal geometry, where scalar curvature, \(Q\)-curvature, and related invariants are governed by conformally covariant differential operators, such as the Yamabe, Paneitz, and GJMS operators. I will discuss how these operators arise naturally in geometric variational problems, including the Yamabe problem and its higher-order analogues, and highlight recent analytical developments. |
Marcos T. O. Pimenta (Universidade Estadual Paulista) Title: Quasilinear elliptic equations involving asymptotically linear operators Abstract: In this work we aim to study in a unified fashion existence and regularity properties of solutions to a very general class of Dirichlet boundary value problems such as $$ \textrm{(P1)} \qquad \begin{cases} – \mbox{div}\left(\displaystyle \phi(|\nabla u|) \nabla u\right) = f(x,u) &\quad \mbox{in } \Omega, \\ u = 0 &\quad \mbox{on } \partial \Omega, \end{cases} $$ where \(\phi:\mathbb{R}^+\to \mathbb{R}^+\) is any function such that \(\phi(s)s\) defines a right-continuous and non-decreasing function which tends to \(1\) as \(s\to+\infty\). Among others, most of the far-famed operators derived from functionals with linear growth, as for instance the \(1\)-Laplacian and the minimal surface operator are included in our model. Under suitable assumptions on the data we prove sharp existence and regularity results for problem \(\textrm{(P1)}\). |
Angela Pistoia (Università di Roma Sapienza) Title: Solutions to elliptic systems in a competitive regime Abstract: I will present some old and new results concerning existence of positive solutions to a class of systems of PDE’s arising in the study of Bose-Einstein condensates in the whole euclidean space in presence of a competitive regime. |
Raoní Ponciano (Universidade Federal do ABC) Title: Trudinger-Moser type inequalities for the Hessian equation with logarithmic weights Abstract: We establish sharp Trudinger–Moser inequalities with logarithmic weights for the \(k\)-Hessian equation and investigate the existence of maximizers. Our analysis extends the classical results of Tian and Wang to \(k\)-admissible function spaces with logarithmic weights, providing a natural complement to the work of Calanchi and Ruf. Our approach relies on transforming the problem into a one-dimensional weighted Sobolev space, where we solve it using various techniques, including some radial lemmas and certain Hardy-type inequalities, which we establish in this paper, as well as a theorem due to Leckband. |
Gaetano Siciliano (Università di Bari) Title: On the Schrödinger-Bopp-Podolsky energy and existence of solutions Abstract: The Bopp-Podolsky theory of electrodynamics is an attempt to overcome the infinity energy problem of the classical Maxwell electrodynamics. In the talk we discuss the natural setting to deal with a Schrödinger equation coupled with the generalized electrodynamic of Bopp-Podolsky in the zero mass case. Then the existence of solution is shown in the subcritical case. Joint work with E. Caponio, P. d’Avenia, A. Pomponio, L. Yang. |
Boyan Sirakov (Pontifícia Universidade Católica do Rio de Janeiro) Title: Uniform a priori estimates for the Lane-Emden system in the plane Abstract: We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. On the other hand, the boundedness may fail if the exponents are not comparable, a surprising incidence of a situation in which the sub-critical Lane-Emden system behaves differently from the scalar equation. Joint work with Nikola Kamburov (PUC-Chile). |
Mayra Soares Costa Rodrigues (Universidade de Brasília) Title: Advances in solving nonlinear Schrödinger equations with general potentials Abstract: We will present some recent results obtained on the existence of a positive solution to the non-linear Schrödinger equation $$-\Delta u +V(x)u=f(u), \quad \text{ in } \mathbb R^N$$ for very general potentials \(V\), with positive or zero limit at infinity (allowing convergence from above, below, or oscillations), but imposing no decay rate assumption. Also, the nonlinearities f may satisfy mild assumptions, including superlinear or asymptotically linear growth at infinity. This is a work in collaboration with Liliane Maia (UnB, Brazil) and Romildo Lima (UFCG, Brazil). |
Michael Struwe (ETH Zürich) Title: The prescribed curvature flow on the disc Abstract: For given functions [/atex]f[/latex] and \(j\) on the disc \(B\) and its boundary \(\partial B=S^1\), we study the existence of conformal metrics \(g=e^{2u}g_{\mathbb{R}^2}\) with prescribed Gauss curvature \(K_g=f\) and boundary geodesic curvature \(k_g=j\). Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz in 2018, we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work in 2005 on the prescribed curvature problem on the sphere, we are able to exhibit a \(2\)-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz by degree-theory. |
Gabriella Tarantello (Università di Roma Tor Vergata) Title: On CMC-immersions of surfaces into Hyperbolic \(3\)-manifolds Abstract: I shall discuss the moduli space of Constant Mean Curvature (CMC) \(c\)-immersions of a closed surface \(S\) (orientable and of genus at least \(2\)) into hyperbolic \(3\)-manifolds. Interestingly when \(|c|<1\), such space can be parametrized by elements of the tangent bundle of the Teichmueller space of \(S\). This is attained by showing that the associated “Donaldson functional” (Gonsalves-Uhlenbeck (2007)) for suitably “constrained” Gauss-Cadazzi equations governing the immersion admits a global minimum as its unique critical point. However, (CMC) \(1\)-immersion into the hyperbolic space play a relevant role in hyperbolic geometry in view of their striking analogies with minimal immersions into the Euclidian space. Hence, I shall discuss the asymptotic behavior of such (CMC) \(c\)-immersions as \(|c|\) approaches \(1\). We see that it is possible to catch at the limit a “regular “ CMC \(1\)-immersions except in very rare situations. For example for genus \(g=2\), we see that “compactness” relates to the image of the Kodaira map on the six Weierstrass points of \(S\). |
Pedro Ubilla (Universidad de Santiago de Chile) Title: The effect of a perturbation on Brezis-Nirenberg’s problem Abstract: We study a class of critical Brézis–Nirenberg-type problems in dimension \(N \geq 3\) for which the classical formulation admits no solution. We show that the introduction of a supercritical perturbation can, nonetheless, lead to the existence of positive solutions. More precisely, we interpret these problems as perturbations of the classical Brézis–Nirenberg equation, where the exponent is modified by a radial term \(r^{\alpha} \), with \( r = |x| \) and \( \alpha \in \left( 0, \min\left\{ \frac{N}{2}, N – 2 \right\} \right) \). This perturbation induces a localized supercritical behavior in the unit ball \( B \subset \mathbb{R}^N \), especially near the origin, where \( r \in (0,1)\). We investigate the influence of this singular supercritical perturbation on the existence of solutions. Remarkably, when \(N = 3\), a novel and unexpected phenomenon emerges: we establish the existence of solutions for certain values of \(\)\lambda\(\) within a range for which the classical Brézis–Nirenberg problem admits none. (To appear in Proc. R. Soc. Edinb., Sect. A.) |
Giusi Vaira (Università di Bari) Title: On the Brezis-Nirenberg problem in low dimensions Abstract: In this talk I will present some recent results regarding the famous Brezis-Nirenberg problem, i.e. a critical problem with a linear perturbation that includes a parameter. I will consider the problem in dimensions N=4,5,6. In particular, I will give results on positive and sign-changing solution as the parameter goes to zero or when the parameter goes to the first eigenvalue of the Laplacian. |