Special Session 66: Geometric insights in Partial Differential Equations: advances and challenges

A three population Lotka-Volterra competition model with two populations interacting through an interface

Pablo Alvarez-Caudevilla Universidad Carlos III de Madrid Spain Co-Author(s):    Cristina Brandle, Monica Molina-Becerra, Antonio Suarez

Abstract: Consider three species competing with each other in the same habitat. One of the species lives in the entire habitat, competing with the other two species, while the other two inhabit two disjoint regions of the habitat. These two populations just interact on a region/interface which acts as a geographical barrier. Under those circumstances we will show how the barrier condition causes a drastic change in species behaviour compared to the classical Lotka-Volterra competitive model.

A critical Kirchhoff type problem in dimension four

Giovanni Anello Department MIFT, University of Messina Italy Co-Author(s):

Abstract: We consider the Dirichlet problem on a bounded smooth domain $\Omega\subset \mathbb{R}^4$, with $0$-boundary data, for the nonlinear critical Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_\Omega|\nabla u|^2dx\right)\Delta u=u^3+\lambda u^{p-1}, \ \ u>0, \ \ {\rm in} \ \ \Omega \end{eqnarray*} where $a,b,\lambda>0$ and $p\in (1,4)$. The main feature of this problem is that the critical exponent for the embedding $W_0^{1,2}(\Omega)\hookrightarrow L^m(\Omega)$ and the exponent of the Kirchhoff term involved in the associated energy functional $I_\lambda$ are both equals to 4. This gives rise to some difficulties in applying variational methods to find solutions. For instance, working with $I_\lambda$, it is not clear how to obtain the boundedness of Palais-Smale sequences or the boundedness of minimizing sequences. We present an approach based on an approximation process that allows to obtain precise constraints on the parameters $b,\lambda$ ensuring the existence of solutions. In particular, we improve previous results where such constraints were not made explicit.

Regularity for (s,p)-harmonic functions

Verena Bogelein University of Salzburg Austria Co-Author(s):    Frank Duzaar, Naian Liao, Giovanni Molica Bisci, and Raffaella Servadei

Abstract: We report on higher Sobolev and H\older regularity results for local weak solutions of the fractional $p$-Laplace equation of order $s\in(0,1)$ with $1

Species that live in different habitats: How do boundary conditions affect their behavior?

Cristina Brandle U. Carlos III de Madrid Spain Co-Author(s):    Pablo \`Alvarez-Caudevilla, M\`onica Molina-Becerra and Antonio Su\`arez

Abstract: In this talk, we will show examples of population models in which two species live in two different regions of the same habitat, separated by a permeable membrane and only come into contact through it. In this context, the membrane acts as a geographical barrier, and the information exchange across it is modeled using the so-called Kedem-Katchalsky conditions. We analyze the existence and uniqueness of positive solutions as a function of the parameters involved in the system.

SYMMETRY CONSTRAINTS AND VARIATIONAL METHODS FOR SCALAR FIELD EQUATIONS ON UNBOUNDED DOMAIN

Giuseppe G DEVILLANOVA Politecnico di Bari Italy Co-Author(s):    Giovanni MOLICA BISCI, Raffaella SERVADEI, Sergio SOLIMINI

Abstract: We consider nonlinear scalar field equations on unbounded domains, where lack of compactness due to translations prevents the direct application of classical variational methods. This difficulty can be overcome by exploiting symmetry. A key idea is to replace symmetry assumptions with suitable symmetry constraints. Building on the planar framework, we use barycentric-type conditions to control the distribution of mass and prevent concentration at infinity. Within this approach, we develop a variational scheme based on constrained min--max constructions, where the constraints encode higher-order symmetry properties of the associated measures. This allows us to recover compactness at the level of Palais--Smale sequences and to obtain nontrivial solutions without imposing radial symmetry. We then analyze the structure of these symmetry constraints, showing how they can be characterized in terms of higher-order barycenters and how, in the planar case, they lead to optimal configurations concentrated on the vertices of regular polygons. We also discuss their extension to higher dimensions, where new geometric and algebraic features arise. This perspective provides a unified framework yielding existence and multiplicity results beyond the classical or block-radial symmetric setting.

Parabolic PDEs with Dynamic Data under a Bounded Slope Condition

Frank Duzaar University of Salzburg Austria Co-Author(s):    Verena Boegelein, Giulia Treu

Abstract: We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with time-dependent boundary data and establish the existence of Lipschitz continuous solutions under minimal assumptions. The main novelty is the introduction of a genuinely time-dependent variant of the classical bounded slope condition. In contrast to existing approaches, this condition allows the supporting hyperplanes of the boundary data to vary in time while remaining uniformly bounded. This yields a flexible geometric framework that accommodates non-stationary boundary values and extends previous results beyond the time-independent setting. Our proof is based on a new barrier construction adapted to the parabolic geometry and the time-dependent boundary behavior. This approach provides control up to the boundary and leads to global Lipschitz bounds for solutions. The method is robust and may be applicable to related parabolic problems. This work was carried out in collaboration with Giulia Treu (University of Padova) and Verena Boegelein (University of Salzburg).

Integrability results for solutions to equations of n--Laplacian type

Luigi D`Onofrio Universita` di Napoli Parthenope Italy Co-Author(s):

Abstract: There is an increasing interest in finding optimal conditions ensuring regularity of solutions to n-Laplacian type equations, so aims of this talk are: to give a complete picture of recent results of integrabilty type in different classes of function spaces and under quite general assumptions and to study the optimality of the estimates found in the setting of Orlicz spaces and in the class of rearrangement invariant spaces

Existence and decay for a Grushin problem in $\mathbb{R}^N$ with singular, convective, critical reaction

Paolo Malanchini Universita` degli Studi di Milano - Bicocca Italy Co-Author(s):    Laura Baldelli, Paolo Malanchini and Simone Secchi

Abstract: We establish an existence result for a problem set in the whole Euclidean space involving the Grushin operator and featuring a critical term perturbed by a singular, convective reaction. Our approach combines variational methods, truncation techniques, and concentration-compactness arguments, together with set-valued analysis and fixed point theory. Additionally, we prove the decay at infinity of solutions in the absence of the convective term. The result is new even in the case where more than one feature between singularity, convectivity and criticality is taken into account. The talk is based on a joint work with Laura Baldelli (Karlsruhe Institute of Technology) and Simone Secchi (Universit\`{a} degli Studi di Milano - Bicocca).

Scaling-based existence and multiplicity results for mixed fractional p-Laplacian equations in R^N

Kanishka Perera Florida Institute of Technology USA Co-Author(s):

Abstract: We present new existence and multiplicity results for the mixed fractional $p$-Laplacian equation \[ {\mathcal A}(u) := (- \Delta)_{p_1}^{s_1}\, u + (- \Delta)_{p_2}^{s_2}\, u = f(|x|,u) \] with \[ u \in {\mathcal D}_\text{rad}^{s_1,p_1}({\mathbb R}^N) \cap {\mathcal D}_\text{rad}^{s_2,p_2}({\mathbb R}^N), \] where $s_i \in (0,1)$ and $1 < p_i < N/s_i$ for $i = 1,2$ and $f$ is a Caratheodory function on $[0,\infty) \times {\mathbb R}$ that satisfies a suitable growth condition. Our results are based on certain scaling properties of the operator ${\mathcal A}$ and recently developed variational methods for scaled functionals.

On a weighted Brezis-Nirenberg-type fractional problem with mixed boundary conditions

Luca Vilasi University of Messina Italy Co-Author(s):    Alejandro Ortega, Youjun Wang

Abstract: We consider an elliptic problem governed by the spectral fractional Laplacian with mixed Dirichlet-Neumann boundary conditions, weighted critical nonlinearities and subcritical perturbations. By using variational arguments we deduce the existence of multiple positive solutions when the weight suitably behaves around its maximum points. In particular, we get different results depending on whether the Dirichlet Sobolev constant is attained or not, Our results extend and improve similar ones obtained in the local case with purely Dirichlet boundary conditions.