Special Session 95: Nonlinear analysis and elliptic boundary value problems

Existence and multiplicity of solutions for different Sturm-Liouville problems

Eleonora Amoroso University of Messina Italy Co-Author(s):

Abstract: In this talk, we present some existence and multiplicity results for Sturm-Liouville problems with either Dirichlet or periodic boundary conditions. Moreover, the problems are parameter-dependent and we provide intervals of parameters for which the problems admit one or multiple solutions. In addition, we offer a link between pure and applied mathematics since the problems considered are used to describe power converters (Dirichlet conditions) or the behavior of a neuron (period conditions). This research is partially supported by PNRR-MAD-2022-12376692 ``AN ARTIFICIAL INTELLIGENCE APPROACH FOR RISK ASSESSMENT AND PREVENTION OF LOW BACK PAIN: TOWARDS PRECISION SPINE CARE`` (XAI-CARE) - CUP: J43C22001510001.

Maximal and minimal weak solutions for elliptic coupled systems with non-linearity on the boundary

Shalmali Bandyopadhyay The University of Tennessee at Martin USA Co-Author(s):    Nsoki Mavinga, Thomas Lewis

Abstract: We analyze weak solutions for a coupled system of elliptic equations with quasimonotone nonlinearity on the boundary. We also formulate a finite difference method to approximate the PDE solutions. We establish the existence of maximal and minimal weak solutions in between ordered pairs of weak sub and supersolutions as well as the existence of maximal and minimal finite difference approximations in between ordered pairs of discrete sub and supersolutions. The analysis employs monotone iteration methods to construct the maximal and minimal solutions when the nonlinearity is monotone. We explore existence, nonexistence, uniqueness and nonuniqueness properties of positive solutions by analyzing particular examples with numerical simulations. When the nonlinearities do not satisfy the monotonicity condition, we prove the existence of weak maximal and minimal solutions using Zorn`s lemma and a version of Kato`s inequality for systems up to the boundary.

Multiplicity of Solutions for a Class of Critical Exponent Problems in the Hyperbolic Space

Mousomi Bhakta Indian Institute of Science Education and Research Pune (IISER Pune) India Co-Author(s):    Debdip Ganguly, Diksha Gupta, Alok Kumar Sahoo

Abstract: In this talk I will discuss the multiplicity of positive solutions to problems of the type $$ -\Delta_{\mathbb B^N} u -\lambda u=a(x) |u|^{2^*-2}u+f(x) \quad\text{in } \mathbb B^N, \quad u\in H^1(\mathbb B^N), $$ where $\mathbb B^N$ denotes the ball model of the hyperbolic space of dimension $N\geq 4$, $2^*=\frac{2N}{N-2}$, $\frac{N(N-2)}{4}<\lambda<\frac{(N-1)^2}{4}$ and $f\in H^{-1}(\mathbb B^N)$ ($f\not\equiv 0$) is a non-negative functional in the dual space of $H^1(\mathbb B^N)$. The potential $a\in L^\infty(\mathbb B^N)$ is assumed to be strictly positive, such that $\lim_{ d(x,0)\to \infty}a(x)=1$, where $d(x,0)$ denotes the geodesic distance. In the profile decomposition of the functional associated with the above equation, concentration occurs along two different profiles, namely, hyperbolic bubbles and localized Aubin-Talenti bubbles. Using the decomposition result, we derive various energy estimates involving the interacting hyperbolic bubbles and hyperbolic bubbles with localized Aubin-Talenti bubbles. Finally, combining these estimates with topological and variational arguments, we establish a multiplicity of positive solutions in the cases: $a\geq 1$ and $a<1$ separately.

Homogenization of a variable exponent problem

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

Abstract: We are concerned with periodic strongly oscillating variable exponent problems involving Leray-Lions type operators. We first show that the problem under study is uniquely solvable and we establish \emph{a priori} estimates. Using a suitable convergence setting, we then provide the homogenization result. This talk is based on a joint work with Renata Bunoiu and Claudia Timofte.

Existence results for a borderline case of a class of p-Laplacian problems

Anna Maria Candela Universita' degli Studi di Bari Aldo Moro Italy Co-Author(s):

Abstract: Let us consider the class of asymptotically ``$p (s + 1)$-linear`` $p$-Laplacian problems \[ \left\{ \begin{array}{ll} - {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right] + s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\ \qquad\qquad\qquad =\ \mu |u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $\Omega$,}\ u = 0 & \hbox{on $\partial\Omega$,} \end{array} \right. \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N \ge 2$, $1 < p < N$, $s > 1/p$, both the coefficients $A_0(x)$ and $A(x)$ are in $L^\infty(\Omega)$ and far away from 0, $\mu \in \mathbb{R}$, and the ``perturbation`` term $g(x,t)$ grows as $|t|^{r-1}$ with $1\le r < p (s + 1)$ and is such that $g(x,t) \approx \nu |t|^{p-2} t$ as $t \to 0$. Under good hypotheses on $g(x,t)$, suitable thresholds for the parameters $\mu$ and $\nu$ exist so that the existence of a nontrivial weak solution of the given problem is proved if either $\nu$ is large enough with $\mu$ small enough or $\nu$ is small enough with $\mu$ large enough. $$ $$ Joint work with Kanishka Perera and Addolorata Salvatore. $$ $$ Partially supported by MUR PRIN 2022 PNRR Research Project P2022YFAJH, Linear and Nonlinear PDEs: New Directions and Applications.

Multiple nonradial solutions of supercritical elliptic problems in exterior domains

Francesca Colasuonno University of Turin Italy Co-Author(s):    Alberto Boscaggin, Benedetta Noris, Tobias Weth

Abstract: In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation \[ -\Delta u + u = a(x)|u|^{p-2}u \] set in an exterior domain of $\mathbb R^N$, $N \ge 3$. Here $p>2$ is allowed to be supercritical in the sense of Sobolev embeddings, and $a$ is a positive weight with additional symmetry and monotonicity properties. In the special case of radial weight $a$, such an existence result ensures the multiplicity of nonradial solutions.

Dissipative gradient nonlinearities prevent blow-up in a class of Keller--Segel models.

Alessandro Columbu Universit\`a degli Studi di Cagliari Italy Co-Author(s):    Tongxing Li; Daniel Acosta Soba; Giuseppe Viglialoro;

Abstract: We examine a class of attraction-repulsion chemotaxis models, which are defined by nonlinearities in the diffusion of cell density, chemosensitivity, and the production rates of chemoattractants and chemorepellents. Additionally, the model includes a logistic term that also depends on the gradient of the biological distribution. In this presentation, we will establish a condition for their boundedness.

Global existence and blow-up lower bounds in a class of tumor-immune cell interactions chemotaxis systems

Rafael Diaz Fuentes University of Cagliari Italy Co-Author(s):    S. Gnanasekaran, A. Columbu, N. Nithyadevi

Abstract: This talk describes the properties of classical solutions to a particular class of chemotaxis systems consisting of three partial differential equations that are either fully parabolic or comprises one parabolic equation along with two others that are elliptic. The primary goal of our investigation is to explore the global existence and potential blow-up of such solutions within bounded domains of $\mathbb{R}^n$, $n \geq 3$, subject to homogeneous Neumann boundary conditions. We establish the global-in-time existence and uniform boundedness of the solutions under smallness conditions imposed on the initial data. Furthermore, we present estimates for the blow-up time of unbounded solutions in three-dimensional space, which are corroborated by numerical simulations.

Nonlocal degenerate variable exponent elliptic problem: existence and multiplicity of solutions

Giuseppe Failla University of Palermo Italy Co-Author(s):    Pasquale Candito, Roberto Livrea

Abstract: In this talk, we will present some recent results on nonlocal $p(x)$ Carrier`s equation with Dirichlet boundary value conditions. One of the main novelties is the possibility of having a sign-changing function in the nonlocal term. To obtain our results, we combine sub-super solution, variational, and truncation techniques. Finally, multiplicity of solutions is obtained via a one-dimensional fixed-point problem.

Dissipation through combinations of nonlocal and gradient nonlinearities in chemotaxis models

Silvia Frassu University of Cagliari Italy Co-Author(s):    Rafael D\`iaz Fuentes, Giuseppe Viglialoro

Abstract: This talk concerns with a class of chemotaxis models in which external sources, comprising nonlocal and gradient-dependent damping reactions, influence the motion of a cell density attracted by a chemical signal. The mechanism of the two densities is studied in bounded and impenetrable regions. In particular, it is seen that no gathering effect for the cells can appear in time provided that the damping impacts are sufficiently strong.

Variational methods for nonlinear differential problems with discontinuous reaction terms

Valeria Morabito University of Messina Italy Co-Author(s):

Abstract: In this talk, we present existence results for a class of nonlinear differential problems with a discontinuous reaction term and explore their link to differential inclusions. The discontinuity of the reaction term necessitates an appropriate framework that can address the nonsmooth behavior of the nonlinearity. To achieve this, our approach is based on variational methods from nonsmooth analysis, specifically focusing on Clarke`s theory of locally Lipschitz functionals.

Mixed finite element methods for fourth order obstacle problems in linearised elasticity

Paolo Piersanti The Chinese University of Hong Kong Shenzhen Peoples Rep of China Co-Author(s):    Tianyu Sun

Abstract: This talk is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence of the error of the numerical approximation of the solution of a biharmonic obstacle problem. The contents of this section are meant to generalise the approach originally proposed by Ciarlet \& Raviart, and then complemented by Ciarlet \& Glowinski. The second problem we consider amounts to studying a two-dimensional variational problem for linearly elastic shallow shells subjected to remaining confined in a prescribed half-space and we show that if the middle surface of the linearly elastic shallow shell under consideration is flat, the symmetry constraint required for formulating the constrained mixed variational problem announced in the second part of the paper is not required, and the solution can thus be approximated by solely resorting to Courant triangles.

Existence of multiple solutions for specific classes of nonlinear anisotropic problems

Angela Sciammetta University of Palermo Italy Co-Author(s):

Abstract: In this talk, we explore several results concerning the existence and multiplicity of solutions to anisotropic Laplacian problems with Dirichlet boundary conditions. Our findings include two nontrivial weak solutions, proven both with and without relying on the Ambrosetti-Rabinowitz condition. Moreover, we point out the existence of three solutions for a particular type of anisotropic Laplacian problem.

On the existence of solutions of degenerate Dirichlet problems with unbounded coefficient

Elisabetta Tornatore University of Palermo Italy Co-Author(s):    G. Bonanno, A. Chinn\`i, B. Di Bella, D. Motreanu

Abstract: This talk is devoted to the study of the existence of solutions of degenerate Dirichlet problems with unbounded coefficient in the principal part. We analyze degenerate problems with convection term and intrinsic operator and degenerate problem with variable exponent. The results presented are part of the research carried out within project PRIN2022-Nonlinear differential problems with applications to real phenomena(20022ZXZTN2).

Multiple critical point results to Sturm-Liouville-type differential problems with highly discontinuous reaction term

Bruno Vassallo University of Palermo Italy Co-Author(s):    Roberto Livrea

Abstract: We consider Sturm-Liouville-type differential problems dependent on a parameter with various boundary conditions. The reaction term $f$ belongs to a class of almost everywhere continuous functions and the set of the points of discontinuity of $f$ may also be uncountable. Combining variational methods with critical point theorems for non-differentiable functions, weak solutions to the problems are established provided the parameter belongs to an explicit interval.

Singular Kirchhoff problems with unbalanced-growth operators

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

Abstract: In this talk, we present existence results for Kirchhoff problems with singular and super-linear reaction terms. The operator is given in a general form, possibly non-homogeneous and with unbalanced growth. Under general assumptions, we prove the existence of two solutions with different energy sign by using the fibering map and the Nehari manifold. Our hypotheses cover the double phase operator and the logarithmic double phase operator as special cases. This is a joint work with Umberto Guarnotta (Ancona).