Special Session 93: Local and nonlocal elliptic boundary value problems

Nonlocal elliptic problems involving the Logarithmic Laplacian

Rakesh Arora Indian Institute of Technology, Varanasi India Co-Author(s):    Jacques Giacomoni and Arshi Vaishnavi

Abstract: This talk presents new sharp continuous and compact embeddings of nonlocal Sobolev spaces (of zero order) in Orlicz spaces and demonstrates the existence of a least-energy weak solution of the Brezis-Nirenberg type, logistic type problem and eigenvalue problem involving the logarithmic Laplacian. For the uniqueness of the solution, we present a new D\`iaz-Saa type inequality, which is of independent interest and can be applied to a larger class of problems.

Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities

Mousomi Bhakta Indian Institute of Science Education and Research Pune (IISER Pune) India Co-Author(s):

Abstract: We will discuss the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian: $$ -\Delta_p u+\varepsilon(-\Delta_p)^s u=\lambda|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }\Omega,\quad u=0 \; \text{ in }\mathbb{R}^N \setminus \Omega, $$ where $\Omega\subset\mathbb{R}^N$ is a bounded open set, $\varepsilon\in(0,1]$, $0

On the homogenization of a class of variable exponent problems

Maria-Magdalena Boureanu University of Craiova Romania Co-Author(s):    Renata Bunoiu, Claudia Timofte

Abstract: We study periodic strongly oscillating variable exponent problems driven by Leray-Lions type operators. We first prove that the problem under consideration admits a unique weak solution and establish a priori estimates. Then, using an appropriate convergence framework, we derive the homogenization result. This talk is based on a joint work with Renata Bunoiu and Claudia Timofte, carried out within Project 27 of the IRN ECO-Math program, Mathematics with Eastern and Central Europe (IMAR, Romania -- CNRS, France).

Weighted Isoperimetric Inequalities for Manifolds with Double Densities and Applications to PDEs

Francesco Chiacchio Universita` degli Studi di Napoli Federico II Italy Co-Author(s):    A. Alvino, F. Brock, A. Mercaldo, C. Nitsch, M. R. Posteraro, and C. Trombetti.

Abstract: We investigate a class of isoperimetric inequalities in R^N characterized by the presence of two different positive weight functions in the volume and surface measures. A central role is played by radial power weights, |x|^a, for which we have solved the isoperimetric problem in many cases. We will present several applications, including Talenti-type estimates for elliptic equations and eigenvalue bounds for Steklov-type problems.

Fourth-order problems on extension domains involving variable exponents

Antonia Chinnì University of Messina Italy Co-Author(s):    M.M.Boureanu and B.Di Bella

Abstract: We present a study on the existence of infinitely many weak solutions for a class of fourth-order elliptic problems involving variable exponents and Navier boundary conditions. The research is characterized by its broad approach, applying the analysis to $W^{2,p(\cdot)}$-extension domains. This category encompasses not only smooth (Lipschitz) domains but also geometrically complex and non-smooth structures, such as fractals (e.g., the Koch snowflake). The problem is driven by nonhomogeneous Leray--Lions type operators, a choice that allows for the simultaneous treatment of several classical operators, including generalized Laplace, mean curvature, and capillarity operators, as well as $p(\cdot)$-biharmonic operators. Furthermore, the study introduces new non-standard operators involving logarithmic and exponential functions. Methodologically, the main result is obtained by applying a variational principle for identifying infinitely many critical points. This approach overcomes the limitations of previous literature in two fundamental ways: absence of the Ambrosetti--Rabinowitz condition and absence of symmetry conditions: The existence of infinitely many solutions is guaranteed without imposing parity or symmetry on the nonlinearity. Instead, the nonlinear term only needs to exhibit suitable oscillatory behavior either at infinity or at zero.

Existence of solutions for a parametric weighted degenerate $p(\cdot)$-Laplacian Dirichlet problem

Beatrice Di Bella University of Messina Italy Co-Author(s):

Abstract: 2

Fractional Dirichlet problems with singular and non-locally convective reaction

Laura Gambera University of Catania Italy Co-Author(s):    Salvatore Angelo Marano

Abstract: In this communication, concerning a work with S. A. Marano, we investigate the existence of positive weak solutions for a Dirichlet problem driven by the fractional $(p,q)$-Laplacian operator. The problem is characterized by a reaction term that combines a weak singularity with a non-local convection dependence, specifically involving the distributional Riesz gradient of the solutions. Due to the presence of the convective term, the problem does not possess a direct variational structure. To address this difficulty, we combine sub-supersolution methods with variational techniques and truncation arguments. Furthermore, fixed-point results are employed to handle the non-local gradient term. Our main result establishes the existence of at least one positive weak solution, extending literature on fractional $(p,q)$ equations to cases involving both singular and non-local gradient terms.

Multiple positive solutions for quasilinear nonlocal problem via topological, variational and set-valued methods

Leszek Gasinski University of the National Education Commission Poland Co-Author(s):

Abstract: We study a nonlocal quasilinear problem driven by the p-Laplacian operator of a nonvariational type, without assuming any kind of monotonicity on the data. The nonlocal term depends on the L^q-norm of the unknown function, where p and q are independent exponents and the weight function can be sign changing. The multiplicity of positive solutions is established through a combination of variational methods, truncation techniques, set-valued analysis, and fixed-point results.

Existence and regularity results for nonlinear fractional order equations

Antonio Iannizzotto University of Cagliari Italy Co-Author(s):    Sunra Mosconi

Abstract: We present some new results dealing with regularity, existence, and multiplicity of weak solutions for elliptic boundary value problems driven by the fractional $p$-Laplacian, and involving several types of reactions both bounded and unbounded. Special emphasis will be placed on boundary H\older regularity, and the ways it affects comparison principles, existence of minimizers, a priori bounds, and multiplicity of solutions as an outcome.

Existence and multiplicity results for differential boundary value problems with possible discontinuous reaction

Roberto Livrea University of Messina Italy Co-Author(s):

Abstract: The aim of the talk is to show some recent results regarding differential problems involving different boundary conditions. In particular, the critical point theory for non-differentiable functions is exploited for studying certain classes of parametric problems. Under suitable assumptions on the reaction terms, the existence of multiple solutions is established provided that the parameter belongs to different computable intervals. References 1. G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1-10 2. P. Candito, R. Livrea and B. Vassallo, Sturm-Liouville equations with discontinuous nonlinearity, preprint 3. R. Livrea, B. Vassallo. Three weak solutions to a periodic boundary Sturm-Liouville problem with discontinuous reaction, Discret. Contin. Dyn. Syst.-Ser. S, 2025, 18(6), 1660-1672. doi: 10.3934/dcdss.2024192

Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities

Artur Marinho Florida Institute of Technology Brazil Co-Author(s):

Abstract: We study a class of nonlinear nonlocal elliptic equations in $R^N$ involving a superposition of Hartree-type nonlinearities. These equations are motivated by the Schr\odinger--Poisson--Slater system and arise as natural generalizations of problems with a single nonlocal interaction term. We consider an equation driven by a family of Riesz potentials weighted by a positive Borel measure, leading to a superposed nonlocal operator. To treat this problem, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb--Sobolev space, and investigate their properties. Using variational methods combined with a recently developed scaling-based critical point theory, we establish existence and multiplicity results for radial solutions for a Brezis--Nirenberg type problem and show that multiplicity depends on the spectral properties of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework to handle superpositions of nonlocal interactions of this type.

On elliptic problems involving local-nonlocal operator under mixed boundary conditions

Tuhina Mukherjee Indian Institute of Technology Jodhpur India Co-Author(s):    Dr. Lovelesh Sharma, Prof. Jacques Giacomoni

Abstract: This talk shall be about a class of elliptic PDEs involving the mixed operator $$\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1)$$ which are prescribed with mixed(Dirichlet-Neumann) boundary conditions. I shall discuss the corresponding eigenvalue problems and bifurcation type result(both from zero and infinity) for an asymptotically linear problem inclined with eigenvalue problems. Firstly, I will provide a functional setup that embeds a variational structure to problems with mixed opeartor and mixed boundary conditions. Under this framework, our first main result was the existence of the first eigenvalue and corresponding positive eigenfunctions with their expected characteristics viz. principal eigenvalue, simple and strictly positive. We also establish boundedness and H\older regularity of eigenfunctions. We also show a strong maximum principle and a few other regularity results which are independent of interest. Secondly, I will show some asymptotic behavior of the first eigenvalues with respect to the Dirichlet set, when Neumann sets dissipate and vice versa. Lastly, I will discuss bifurcation type results, built upon the strong maximum principle and H\older regularity results, for an asymptotically linear problem inclined with the eigenvalue problems.

Variational analysis of elliptic hemivariational inequalities with nonmonotone boundary conditions

Anna Ochal Jagiellonian University in Krakow Poland Co-Author(s):

Abstract: We investigate a class of nonlinear elliptic boundary value problems formulated as hemivariational inequalities. These problems arise naturally in the study of steady-state heat conduction where the boundary conditions are nonmonotone and multi-valued. Such conditions are described by the Clarke generalized gradient of a locally Lipschitz continuous function on a portion of the boundary. Applying variational methods, we establish the existence of solutions for this class of inequalities [1]. In this talk, we discuss the system`s sensitivity to parameters such as the heat transfer coefficient. We present results concerning the existence of optimal solutions for related control problems and provide an asymptotic analysis showing the behavior of the system states as the parameter tends to infinity. This contribution aligns with new trends in nonlinear boundary value problems by showcasing the application of subdifferential analysis to local differential operators with complex boundary interactions. [1] C.M. Gariboldi, S. Migorski, A. Ochal, and D.A. Tarzia, Existence, comparison, and convergence results for a class of elliptic hemivariational inequalities, Appl. Math. Optim., 84 (Suppl 2) (2021), S1453--S1475.

Convergence Analysis of a Differential Variational Inequality

Mircea Sofonea University of Perpignan France Co-Author(s):    Mircea Sofonea and Bruno Vassallo

Abstract: We deal with the study of a special class of systems which couple an implicit differential equation with an evolutionary variational inequality. Following the terminology used in the literature, we refer to such systems as differential variational inequalities (DVI). Inequalities of this form arise in a large number of mathematical models in Solid and Contact Mechanics. We start with a one dimensional rheological example which leads to a DVI. Inspired by this example, we formulate the problem in the abstract framework of a real Hilbert space, then we state and prove an existence and uniqueness result. The proof is based on a result of evolutionary variational inequalities combined with a fixed point argument for history-dependent operators. We proceed with a convergece criterion, that is, we identify necessary and sufficient condition which guarantee the convergence of an arbitrary sequence to the solution. We introduce a well-posedness concept for the DVI we study, then and prove the corresponding well-posedness result. Finally, we apply these abstract results in the study of the one-dimensional rheological model and provide the corresponding mechanical interpretations.

The superposition of operators of mixed fractional order and their applications

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

Abstract: In this talk we introduce the superposition operator \begin{equation}\label{superposition} \int_{[0, 1]} (-\Delta)^s u \, d\mu(s), \end{equation} where $\mu$ denotes a (Borel) signed measure on the fractional interval $[0, 1]$. We discuss the existence of solutions for some problems in presence of the operator \eqref{superposition} and subject to different types of boundary conditions. These results are part of some joint works with Serena Dipierro, Enrico Valdinoci and Kanishka Perera.

Morse-theoretic approach for logarithmic double phase problems

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

Abstract: In this talk, we investigate nonlinear elliptic problems involving logarithmic double phase operators and superlinear nonlinearities. Using critical group theory and variational methods, we show that the problem admits at least three distinct nontrivial bounded weak solutions. Two of these solutions have constant sign and positive energy, while a third is obtained via a topological argument involving the structure of the critical groups at zero and at infinity. This talk is based on joint works with Franziska Borer (Berlin), Vicen\c{t}iu D. R\u{a}dulescu (Krakow) and Matheus F. Stapenhorst (Presidente Prudente).