Special Session 35: Elliptic PDEs: singularities, discontinuities, and nonlinear phenomena

The nonlinear fractional relativistic Schrodinger-Choquard equation

Vincenzo Ambrosio Universita` Politecnica delle Marche Italy Co-Author(s):

Abstract: In this talk, I will consider the doubly nonlocal nonlinear elliptic equation $(-\Delta+m^{2})^{s}u+\omega u=(I_{\alpha}*F(u)) F`(u)$ in $\mathbb{R}^{N}$, where $N\geq 2$, $s\in (0, 1)$, $m>0$, and $\omega>-m^{2s}$. Here, $(-\Delta+m^{2})^{s}$ denotes the fractional relativistic Schrodinger operator, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0, N)$, and $F:\mathbb{R}\to \mathbb{R}$ is a $C^1$-nonlinearity of Berestycki--Lions type. I will discuss the existence of least energy solutions, as well as their qualitative properties, including regularity, decay, sign, and symmetry.

Dirichlet boundary value problems governed by non-monotone differential operators

Francesca Anceschi Universita` Politecnica delle Marche Italy Co-Author(s):

Abstract: We prove existence results for Dirichlet boundary value problems for nonlinear possibly singular equations involving a generic possibly non-monotone differential operator defined in a open interval J. Also, under very mild assumptions, we prove the existence of solutions for suitably prescribed boundary conditions, and we also address the study of the existence of heteroclinic solutions on the half-line. These results are part of a joint work with C. Marcelli and F. Papalini.

Gradient regularity for quasilinear elliptic problems

Carlo Alberto Antonini INdAM (National Institute of High Mathematics) Italy Co-Author(s):

Abstract: In this talk we will deal with quasilinear elliptic equations in divergence form $-\mathrm{div}\big(\mathcal{A}(x,Du)\big)=f$, modeled upon the $p$-Laplace and, more generally, the Orlicz-Laplace operator. We will discuss gradient regularity of solutions, both in the interior and up to the boundary, under Dirichlet or Neumann boundary conditions.

Critical double phase Kirchhoff problems in $\mathbb{R}^N$

Giuseppina Autuori Polytechnic University of Marche Italy Co-Author(s):    Teresa Isernia

Abstract: In this talk we will discuss the existence of groundstate solutions for certain critical Kirchhoff problems governed by a double phase operator in $\mathbb{R}^N$, involving a bounded periodic potential, a positive weight and a reaction term. Under suitable growth and monotonicity assumptions, we prove an existence result according to the size of a positive parameter $\lambda$, without the {\it Ambrosetti--Rabinowitz} condition, in the setting of Musielak--Orlicz spaces. Our proof technique relies on variational arguments including the Mountain Pass Theorem, the Nehari manifold method, concentration compactness results and the use of a suitable limiting problem. The talk is based on a joint work with {\it Teresa Isernia}.

The obstacle problem for the $p$-Laplacian

Annamaria Barbagallo University of Naples Federico II Italy Co-Author(s):    Umberto Guarnotta

Abstract: The purpose of the talk is to prove the existence of a solution to the obstacle problem involving the $p$-Laplacian operator, in a setting where the reaction term is singular at zero and exhibits strong discontinuities. Notably, no restriction is placed on the discontinuity set having zero Lebesgue measure. Lastly, it is shown that the solution lies in the space $C^{1, \alpha}$ outside the contact set.

Boundedness of solutions to Dirichlet, Neumann, Robin and mixed problems for elliptic equations in Orlicz spaces

Giuseppina G. Barletta University of Reggio Calabria Italy Co-Author(s):    A. Cianchi, G. Marino, E. Tornatore

Abstract: We present some boundedness results for boundary value problems for second-order elliptic equations in divergence form. We examine different boundary conditions and higlight the decisive role played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

Quasi-variational inequalities with non compact constraints

Irene Benedetti Department of Mathematics and Computer Science, University of Perugia Italy Co-Author(s):

Abstract: This talk deals with generalized quasi-variational inequalities, namely problems involving a multivalued operator and a constraint set depending on the solution itself. These models arise in several applications, including equilibrium theory, optimization, and problems related to elliptic equations. We present new existence results obtained under weaker assumptions than those typically required in the literature. In particular, we relax both the compactness conditions on the constraint set and the regularity assumptions on the multivalued operator. The approach is based on tools from nonlinear functional analysis and fixed point theory. Our results extend the classical framework and provide a more flexible setting for the analysis of quasi-variational inequalities.

Some recent results on singular anisotropic elliptic equations

Barbara Brandolini Universita` degli Studi di Palermo Italy Co-Author(s):

Abstract: In this talk, we discuss some recent results on the existence and uniform boundedness of solutions for a general class of Dirichlet anisotropic elliptic problems, defined in open bounded subsets of $\mathbb{R}^N$ $(N\ge 2)$, and driven by the the so-called $\overrightarrow{p}$-Laplacian operator, where $\overrightarrow{p}=\left(p_1,p_2,\dots,p_N\right)$, with $N/p=\sum_{k=1}^N (1/p_k)>1$. The feature of this study is the inclusion of a possibly singular gradient-dependent term $\Psi(u,\nabla u)=\sum_{j=1}^N |u|^{\theta_j-2}u\, |\partial_j u|^{q_j}$, where $\theta_j>0$ and $0\leq q_j < p_j$ for $1\leq j\leq N$. We also investigate the case $q_j=p_i$ for every $1\leq j\leq N$ in the singularity-free case, and we prove the existence of a solution which a further regularity of exponential type. Based on joint works with Florica C. C\^irstea (The University of Sydney, Australia) and V. Ferone (Universit\`a degli Studi di Napoli Federico II).

On the weak Harnack inequality for a generalized Orlicz De Giorgi class

simone ciani University of Bologna Alma Mater Italy Co-Author(s):    Eurica Henriques, Igor I. Skrypnik

Abstract: We introduce a generalization of De Giorgi classes with Orlicz growth. The aim is twofold: on the one hand, to encompass a broader class of functionals and equations; on the other hand, to provide a definition based on a simple energy inequality that does not rely on an underlying functional, but rather on the intrinsic scaling between radii and levels. For these classes, we establish a Weak Harnack inequality, thereby unifying its validity across non-uniformly elliptic equations, double-phase and degenerate double-phase functionals, as well as functionals with variable exponents.

Comparison results for the fractional heat equation with a singular lower order term

Ida de Bonis Sapienza University of Rome Italy Co-Author(s):

Abstract: We provide symmetrization results in the form of mass concentration comparisons for fractional singular parabolic equations in infinite cylinders of the type $\Omega\times (0,T)$, where $\Omega\subset \mathbb{R}^N, N\geq 2$ is a bounded, open set with Lipschitz boundary, and $T>0$.\ The results are obtained in collaboration with B. Brandolini, V. Ferone and B. Volzone.

Asymptotic behavior as $p\\to 1$ of solutions to $p$-Laplacian problems in presence of convection and drift terms

Riccardo Durastanti Universita degli Studi di Napoli Federico II Italy Co-Author(s):

Abstract: We deal with the asymptotic behavior as $p\to 1^+$ of weak solutions to nonlinear Dirichlet problems modeled by \begin{equation*} \begin{cases} -\Delta_p u + |u|^{p-1}u = -\text{\text{div}}\left(|u|^{p-2}u E(x)\right) + |\nabla u|^{p-2}\nabla u\cdot F(x) + f(x) & \text{ in }\Omega, \ u=0 & \text{ on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge 2$). We focus on the a priori estimates depending on the norm of the Lebesgue functions $E,F$ and $f$ which guarantee the existence of a limit problem in the framework of functions of bounded variation. This is a joint work with Francescantonio Oliva.

First eigenvalue and torsional rigidity: isoperimetric inequalities for the fractional Laplacian

Vincenzo Ferone Universit\`a di Napoli Federico II Italy Co-Author(s):

Abstract: We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset R^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse H\older inequality for an eigenfunction corresponding to $\lambda_1(\Omega)$. The results have been obtained in collaboration with B. Brandolini, I. de Bonis, G. Piscitelli and B. Volzone.

On Bobkov-Tanaka type spectrum for the double-phase operator

Laura Gambera University of Catania Italy Co-Author(s):    Umberto Guarnotta

Abstract: In this talk, based on joint work with U. Guarnotta, we study an eigenvalue problem for a double-phase operator. The lack of homogeneity of $\Delta_p^a + \Delta_q$ makes the notion of spectrum non-unique, allowing for different possible definitions, as already occurs for the $(p,q)$-Laplacian. We adopt a notion of spectrum inspired by the Fucik spectrum, following an approach previously introduced by Bobkov and Tanaka. A key point is that the structure of the spectrum depends strongly on the linear independence condition $\phi_p^{a} \neq k \phi_q$ for every $k \in \mathbb{R}$, leading to different spectral configurations depending on whether this condition holds. The problem exhibits several features, including the lack of homogeneity, unbalanced growth placing it in the Musielak-Orlicz framework, and the presence of a non-negative, non-trivial weight $a \in C^{0,1}(\Omega)$, which leads to a loss of local regularity. We discuss the region of parameters in which existence or non-existence of positive solutions occurs. The analysis relies on normalization arguments, the Nehari manifold, and truncation techniques, combined with Picone-type inequalities and a suitably tailored strong maximum principle.

Normalized solutions for Elliptic PDEs

Divya Goel Indian institute of Technology BHU India Co-Author(s):

Abstract: Normalized solutions for elliptic PDEs are sought under a prescribed mass constraint and appear naturally in models from nonlinear physics. In this talk, we will discuss recent results for normalized solutions to nonlinear elliptic problems involving Kirchhoff, Choquard, fractional, and weighted Caffarelli-Kohn-Nirenberg type operators. The focus will be on variational methods under mass constraint, where critical growth, nonlocal effects, and loss of compactness create significant difficulties. Using Pohozaev-type identities, fibering maps, and minimax arguments, one can establish existence, multiplicity, and ground state solutions in several regimes. These results demonstrate that constrained variational techniques provide a unified framework for studying normalized solutions in modern elliptic PDEs.

Multiple positive and negative energy solutions for $(p, q)$--Kirchhoff critical equations in $\mathbb{R}^{N}$

Teresa Isernia Universita` Politecnica delle Marche Italy Co-Author(s):    Giuseppina Autuori and Letizia Temperini

Abstract: In this talk, we deal with a class of nonlocal $(p, q)$-Kirchhoff equations in $\mathbb{R}^{N}$ involving competing nonlinearities and a critical growth term. The interplay between the nonlocal Kirchhoff coefficient and the combined action of subcritical and critical nonlinearities leads to a rich variational structure, making the analysis particularly delicate. We focus on the existence and multiplicity of solutions, highlighting how the behavior of the system strongly depends on the interplay between the exponents and the parameter $\lambda$. In particular, we identify different regimes in which multiple solutions arise, corresponding to negative and positive energy levels, and driven by either small or large values of $\lambda$, as well as suitable conditions on the weight functions. Our results significantly extend and refine existing results on $(p, q)$-Kirchhoff problems with critical growth. This talk is based on a joint work with Giuseppina Autuori and Letizia Temperini.

Recent advances in the regularity of some classes of elliptic systems of PDEs

Salvatore Leonardi University of Catania Italy Co-Author(s):

Abstract: We present an overview of several regularity results for specific classes of elliptic systems established in recent years. Specifically, employing the Einstein summation convention, we investigate the following system: \[ \left\{\begin{aligned} &u\in\W\\ & -\Di \, A_i^\nu(x, u, \DD u)=-\Di\big(E_{i}(x) \, u^{\nu}\big)+f^\nu(x), \quad \text{$x\in\Om$}, \, \nu=1, \ldots N \end{aligned}\right. \] defined on a domain $\Om \subset \R^n$ ($n \geq 3$, $N \geq 2$), where $A_i^{\nu}$ and $E_i(x)$ are required to satisfy appropriate structural and integrability conditions. The analysis further assumes that the data $f^{\nu}$ belong to the Lebesgue space $L^{t}(\Om)$ for $\nu=1, \ldots, N$, with $t \geq \dfrac{2n}{n+2}$.

Remarks on maximum principle

Francesco Leonetti University of L`Aquila Italy Co-Author(s):

Abstract: Maximum principle can be obtained for solutions to elliptic equations. De Giorgi`s counterexample says that this is no longer true for systems, but, despite that, it is possible to get maximum principle for solutions to some systems.

Multiplicity of mountain-pass type transition solutions for Lagrangian systems with double-well potentials

Piero Montecchiari Universita` Politecnica delle Marche Italy Co-Author(s):

Abstract: We present several results concerning the multiplicity of heteroclinic and homoclinic solutions of mountain-pass type for non-autonomous Lagrangian systems of Allen-Cahn type. In particular, we show that, under natural non-degeneracy assumptions, multi-transition local minimizers generate distinct local mountain-pass structures, giving rise to multiple associated mountain-pass type solutions.

1-Laplace Dirichlet problems involving first order terms

Francescantonio Oliva Sapienza university of Rome Italy Co-Author(s):

Abstract: In this talk, we discuss the existence and qualitative behavior of bounded variation (BV) solutions to Dirichlet problems driven by the 1-Laplace operator. We deal with general, and possibly singular, first-order nonlinearities and we focus on how these nonlinearities play a regularizing effect on the solutions. In particular, we study the role of the gradient nonlinearity in the attainment of the homogeneous boundary condition for the solutions.

Scaling-based existence and multiplicity results for double phase problems in R^N

Kanishka Perera Florida Institute of Technology USA Co-Author(s):    Francesca Colasuonno, Patrick Winkert

Abstract: \usepackage{amsmath,amssymb} We present new existence and multiplicity results for the double phase problem \[ {\mathcal A}(u) := - \text{ div} \left(|\nabla u|^{p-2}\, \nabla u + |x|^{-b}\, |\nabla u|^{q-2}\, \nabla u\right) = f(|x|,u) \] with \[ u \in {\mathcal D}_\text{rad}^{1,p}({\mathbb R}^N) \cap {\mathcal D}_\text{rad}^{1,q}({\mathbb R}^N;|x|^{-b}), \] where $1 < p < q < N$, $0 \le b < N - q$, and $f$ is a Carath\`{e}odory 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.

Optimal Global $BV$ Regularity for 1-Laplace type BVP`s with Singular Lower Order Terms

Francesco Petitta Sapienza, University of Rome Italy Co-Author(s):

Abstract: This talk investigates the optimal regularity for solutions to singular elliptic equations involving the 1-Laplace operator. We analyze a class of problems whose model case is $-\Delta_1 u = f(x)u^{-\gamma}$ with homogeneous Dirichlet boundary conditions. In contrast to the classical $p$-Laplacian case with $p>1$, we prove that the structure of the $1$-Laplacian guarantees global $BV$ regularity of the solutions for any power $\gamma > 0$. We will show how a priori estimates based on truncation arguments allow us to handle the boundary discontinuity issue and to control the effects arising from the domain's curvature.

Normalised solutions to a fractional Schr\\{o}dinger equation in the strongly sublinear regime

Jacopo Schino University of Warsaw Poland Co-Author(s):    Marco Gallo

Abstract: Schr\{o}dinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties. One of them is the conservation of mass, which gives rise to the search for normalised solutions. In this talk, I will explain a possible approach to solve a fractional equation paired with a constraint on the $L^2$ norm. The context includes the so-called strongly sublinear regime, i.e., when the RHS has a negative sublinear growth at the origin. This makes a direct variational approach impossible because the energy functional is not well-defined in $H^s(\mathbb{R}^N)$. In the proposed approach, when the mass is sufficiently large, a family of approximating problems is considered so that the energy functional is of class $\mathcal{C}^1$ and a corresponding family of solutions is obtained, which eventually converge to a solution to the original problem. In the strongly sublinear regime, the previous result for a suitably translated problem is exploited to obtain a solution for any mass.

Nonlocal operators in divergence form and existence results for $L^1$ data

Caterina Sportelli Universidad de Granada Spain Co-Author(s):

Abstract: In this talk, we study the existence and uniqueness of weak solutions to Dirichlet boundary value problems driven by a nonlocal operator in divergence form with data in $L^1(\Omega)$ that are suitably dominated. Moreover, we show that, as $s \nearrow 1$, the nonlocal solutions converge to the solution of the corresponding local problem, thereby recovering the classical theory as a limit case. The results presented in this talk are based on \cite{MR5007887}. \begin{thebibliography}{99} \bib{MR5007887} Arcoya, David, Dipierro, Serena, Proietti Lippi, Edoardo, Sportelli, Caterina, Valdinoci, Enrico, Nonlocal operators in divergence form and existence theory for integrable data, J. Funct. Anal., 290, 2026, 7, Paper No. 111317, 58, issn 0022-1236, MR{5007887}, doi{10.1016/j.jfa.2025.111317}, \end{thebibliography}

Existence results for logarithmic double phase problems with nonlinear Neumann boundary conditions

Patrick Winkert University of Technology Berlin Germany Co-Author(s):

Abstract: In this talk, we present several existence results for logarithmic double phase problems with nonlinear Neumann boundary conditions. By introducing a new and very general equivalent norm in the logarithmic Musielak-Orlicz Sobolev space, we discuss different types of nonlinearities on the boundary, including critical growth. This talk is based on joint works with Franziska Borer (Berlin), Yino B. Cueva Carranza (Presidente Prudente), Leszek Gasi\`{n}ski (Krakow), Marcos T.O. Pimenta (Presidente Prudente), Eylem \{O}zt\{u}rk (Ankara) and Matheus F. Stapenhorst (Presidente Prudente).