M. ABDELAZIZ Hellal

We prove the existence of weak solutions $(u, v) \in H_0^1(\Omega) \times H_0^1(\Omega)$ for a doubly singular elliptic system patterned on the Schr\"odinger--Maxwell framework. Under the assumption that the reaction term involves a datum $f \in L^{m(\cdot)}(\Omega)$ with $m : \overline{\Omega} \to (1,\infty)$ continuous, we establish positive solutions for appropriate ranges of $m(x)$. Our results improve upon the theory for single singular equations, and we show that integrability conditions on $f$ yield higher integrability of the solutions via approximation schemes and a priori estimates.

In this paper, we investigate the existence and regularity of nonnegative weak solutions for specific class of nonlinear singular anisotropic elliptic problems with degenerate coercivity involving variable exponents. We show that certain lower-order term contribute to the regularization of solutions, as well as the regularization induced by a singular nonlinearity term. Our methodology employs an approximation technique that integrates anisotropic variable exponent Sobolev spaces, truncation methods, compactness arguments, and Schauder’s fixed-point theorem. These findings extend some previous results established under constant exponents.

This work investigates a class of nonlinear elliptic problems posed on a bounded domain \(\Omega \subseteq \mathbb{R}^N\) (\(N \geq2\)), described by the partial differential equation
\[
-\mathrm{div}\left(H(x,\nabla Z)\right)=F,
\]
where \(H\) is an operator of Leray-Lions type acting from the space \( W^{1,\alpha(\cdot)}_0(\mathcal{A}) \) into its dual. When the right-hand side \(F\) belongs to \( L^{\theta(\cdot)(\mathcal{A})\), with \(\theta(\cdot)> 1 \) satisfying certain conditions, we prove the existence of weak solutions for this class of problems under \(\alpha(\cdot) \)-growth conditions. Our approach is based on a combination of variational methods, approximation techniques, and compactness arguments. The functional framework involves Sobolev spaces with variable exponents, \(W^{1,\alpha(\cdot)}_0(\mathcal{A})\), as well as Lebesgue spaces with variable exponents, \(L^{\alpha(\cdot)}(\mathcal{A}) \).

This course is designed for second class license mathematics academic (LMD) students for the third semester, introducing the fundamental concepts of topology necessary for
advanced mathematical study. This handout is the result of ten academic years of teaching experience in the Department of Mathematics at the University of M’sila. It is composed of four chapters. At the end of each chapter, there are exercises, with solutions provided at the end of the course. Additionally, some exams with their keys answers are included. The first chapter presents preliminary concepts on metric spaces. The second chapter explores the structure of topological spaces and their properties. The third chapter covers compactness and connectedness. Finally, the fourth chapter introduces the concepts and properties of normed vector spaces and continuous linear applications, along with key theorems commonly used in
functional analysis and PDEs, including their applications.

Certificate of Appreciation

This is to certify that
Hellal Abdelaziz, University of M'sila

Served as the President of the Organizing Committee for the
9th M'sila Conference on Mathematical Analysis,
held in M'sila on November 12-13, 2025.

We extend our sincere thanks for his valuable contributions and expertise.

In work, we demonstrate a trace theorem that facilitates the handling of Neumann problems with nonlinear boundary terms in anisotropic spaces characterized by variable exponents. Following this, we explore a problem involving generalized p(⋅)-Laplacian-type operators. We establish the existence of solutions and pay particular attention to cases where the solution is unique.

This work examines a class of double-phase elliptic problems involving variable exponents and irregular data. Under carefully chosen assumptions on the data, we apply the compactness method to prove the existence of weak solutions to the problem. Moreover, we explore the regularizing effect introduced by the additional term, which enhances the summability properties of the solution for the double-phase problem being studied. Our analysis provides deeper insights into the interplay between the structure of the problem and the regularity of its solutions.

I am pleased to acknowledge that I was awarded a Certificate of Recognition for my contribution to the National Conference on Mathematical Analysis organized by the University of M'sila.

This work investigates the regularizing effects in a class of anisotropic nonlinear Dirichlet problems. We consider elliptic equations with anisotropic growth conditions, where the nonlinearity and the associated differential operator exhibit non-standard behavior influenced by directional dependencies. Under suitable assumptions on the structure of the problem, we establish the existence and uniqueness of weak solutions. Furthermore, we analyze the regularizing effects induced by the anisotropic nature of the problem, which lead to improved regularity and summability properties of the solutions. Our results highlight the interplay between the anisotropic structure and the smoothing mechanisms, providing a deeper understanding of how directional growth conditions influence the qualitative behavior of solutions. These findings extend existing theories and offer new insights into the analysis of anisotropic partial differential equations.

This study focuses on a category of double-phase elliptic problems characterized by variable exponents and irregular data. By imposing appropriate conditions on the data, we utilize the compactness approach to demonstrate the existence of weak solutions to the problem. Furthermore, we emphasize the regularizing influence of the additional term on the summability properties of the solution for the double-phase problem under investigation.

This presentation will focus on some existence results for double phase problems.

The aim of this presentation is to study a Dirichlet problem characterized by degenerate coercivity, involving variable exponents and a singular nonlinearity

The main focus of this presentation is singular anisotropic degenerate elliptic equations with variable nonlinearities.

The presentation was focused on How can I choose my specialty?

Science, Technology, Engineering, and Mathematics, considered as a group of academic or career fields (often used attributively). We are interested in the challenges and the perspectives of teaching STEM in the technical classroom, so what are challenges and perspectives of teaching STEM in the technical classroom?
The aim of this presentation is how to explain and clarify some formulas via geometrical method using some technical classroom.
N.B. We refer the link: https://www.mathdoubts.com/a-plus-b-whole-square-geometric-proof/

Abdelaziz, H. (2023). Singular Elliptic Equations with Variable Exponents. International Journal of Mathematics And Its Applications, 11(4), 141–168. Retrieved from https://ijmaa.in/index.php/ijmaa/article/view/1440

The main point of this presentation is studying the existence and regularity of solutions for some nonlinear parabolic equations with a principal part characterised by degenerate coercivity involving varaible exponents.

In our work, we present the prove of the existence and the regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with pi(·) growth conditions, degenerate coercivity and Lm(·) data, with m(·) > 1 being small. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

In this work we prove the existence and uniqueness of both entropy solutions and renormalized solutions for an anisotropic parabolic p-Laplacian equations with $p_i$-growth conditions using the penalization method. Moreover, we obtain that entropy solutions coincide with the renormalized solutions.

In this paper, we prove the existence and the regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with pi( ⋅ ) growth conditions, degenerate coercivity and L^m( ⋅ ) data, with m( ⋅ ) > 1 being small. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

This paper is concerned with the study of the nonlinear elliptic equations in a bounded subset Ω ⊂ R^N
Au = f,
where A is an operator of Leray-Lions type acted from the space W^{1,p(·)}_{0}(Ω) into its dual. when the second term f belongs to L^m(·), with m(·) > 1 being small. we prove existence and regularity of weak solutions for this class of problems p(x)-growth conditions. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents.

In this work we prove the existence and uniqueness of both entropy solutions and renormalized solutions for an anisotropic parabolic $\overrightarrow{p}$-Laplacian equations with $p_i$-growth
conditions using the penalization method. Moreover, we obtain that entropy solutions coincide with the renormalized solutions.

This paper is concerned with study of the nonlinear singular elliptic equations in a bounded domain $\Omega\subset\mathbb{R}^N$, $(N\geq 2)$ with Lipschitz boundary $\partial\Omega$,
$$
-\mathrm{div}\left(\widehat{a}(\cdot,Du)\right)=\frac{f}{u^{\gamma(\cdot)}},
$$
Where $f$ is a nonnegative function belonging to the Lebesgue space with variable exponents $L^{m(\cdot)}(\Omega)$, with $m(\cdot)$ being small (or $L^{1}(\Omega)$), while $m:\overline{\Omega}\to (1,+\infty)$,
$\gamma:\overline{\Omega}\to (0,1)$ are continuous functions satisfying certain conditions depend on $p(\cdot)$. We prove the existence, uniqueness and regularity of nonnegative weak solutions for this class of problems with $p(\cdot)$-growth condition. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents. Our results canbe seen as a generalization of some results given in the constant exponents case.

I contributed greatly to success the 7th M’sila Conference on Mathematical Analysis (7th MCMA 2022) as a Member of the organizing committee.

In this poster, we show you how to prove the existence and the regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with p i (·) growth conditions, degenerate coercivity and L^m(·)data, with m(·) > 1 being small. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

In this work, we prove existence and regularity of weak solutions for a class of p(·)-Laplacian equations with variable exponents and L m data, with m being small.
The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

In this work, we prove existence and regularity of weak solutions for a class of p(x)-Laplacian equations with variable exponents and L^m data, with m being small. The functional setting involves Lebesgue−Sobolev spaces with variable exponents. The study of our problem in [4] is a new and interesting topic. Inspired by [1], [2] and [3] we prove the existence of weak solution for our problem with right-hand side in L^m and the variable exponent p(x) satisfies a condition of regularity. The main steps of the proof consist of obtaining uniform estimates for suitable approximate problems and then passing to the limit.

[1] M. Bendahmane, P. Wittbold, Renormalized solutions for nonlinear elliptic equations with
variable exponents and L^1− data, Nonlinear Analysis TMA 70(2), 567-583, (2009).
[2] L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm.
Partial Differential Equations 17, 641-655, (1992).
[3] L. Diening, P. Harjulehto, P. Hasto, M. Rauzicka; Lebesgue and Sobolev Spaces with Variable
Exponent, Book, Lectures Notes in Mathematics 2017. Springer-Verlang Berlin Heidelberg, (2011).
[4] F. Mokhtari, K. Bachouche, H. Abdelaziz, Nonlinear elliptic equations with variable exponents
and measure or L^m data. J. Math. Sci. 35, 73-101, (2015).

In this work, we prove existence and regularity of weak solutions for a class of p(·)-Laplacian equations with variable exponents and Lm data, with m being small. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

Abstract. In this work, we prove existence and regularity of weak solutions for a class of p(·)-Laplacian equations with variable exponents and L^m data, with m being small.
The functionalsetting involves Lebesgue-Sobolev spaces with variable exponents.

In this paper we prove existence and regularity of weak solutions for a class of nonlinear elliptic equations with variable exponents. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

In this work, we prove existence and regularity of weak solutions for a class of nonlinear elliptic equations with variable exponents and measure or L^m data, with m being small. The functional setting involves Lebesgue-Sobolev spaces with variable exponents.

University of Sciences and Technology Houari Boumediene
BP 32 EL ALIA, Bab Ezzouar, Algiers, 16111, Algeria

Hellal Abdelaziz
Attestation de participation
Je soussigné, Abdelhafid Mokrane,
Président du Comité d'Organisation des Premières Journées Scientifiques du Laboratoire Euro Maghrébin de Mathématiques
et de leurs Interactions (LEM2I),
a participé aux activités de ces Journées, qui se sont tenues du 13 au 22 juin 2010, au Centre Munatec de Tipaza, Algérie.
certifie que
Ministère de l'Enseignement Supérieur et de la Recherche Scientifique
Ecole Normale Supérieure de Kouba – Alger
ENS Kouba
Premières Journées Scientifiques du Laboratoire Euro Maghrébin de Mathématiques et de leurs Interactions (LEM2I)
13-22 juin 2010, Tipaza, Algérie