arXiv subjects

Ala Eddine Bahrouni

Publications and source records attributed to Ala Eddine Bahrouni.

3 recordsLinked to original sources

arXiv2025

The concentration-compactness principle for Musielak-Orlicz spaces and applications

This paper extends the Concentration-Compactness Principle to Musielak-Orlicz spaces, working in both bounded and unbounded domains. We show that our results include important special cases like classical Orlicz spaces, variable exponent spaces, double phase spaces, and a new type of double phase problem where the exponents depend on the solution. Using these general results with variational methods, we prove that certain quasilinear equations with critical nonlinear terms have solutions.

math.AP↗

arXiv2025

A new class of anisotropic double phase problems: exponents depending on solutions and their gradients

In this work, we introduce two novel classes of quasilinear elliptic equations, each driven by the double phase operator with variable exponents. The first class features a new double phase equation where exponents depend on the gradient of the solution. We delve into proving various properties of the corresponding Musielak-Orlicz Sobolev spaces, including the $Δ_2$ property, uniform convexity, density and compact embedding. Additionally, we explore the characteristics of the new double phase operator, such as continuity, strict monotonicity, and the (S$_+$)-property. Employing both variational and nonvariational methods, we establish the existence of solutions for this inaugural class of double phase equations. In the second category, the treatment of exponents is dependent on the solution itself. This class differs from the first one due to the unavailability of suitable Musielak-Orlicz Sobolev spaces. For this reason, we employ a perturbation argument that leads to the classical double phase class. These two new classes highlight how different physical processes like the movement of special fluids through porous materials, phase changes, and fluid dynamics interact with each other. Our results are novel in this context and includes a self-contained techniques.

math.AP↗

arXiv2024

Double phase problems with variable exponents depending on the solution and the gradient in the whole space $\mathbb{R}^N$

In this paper, we establish continuous and compact embeddings for a new class of Musielak-Orlicz Sobolev spaces in unbounded domains driven by a double phase operator with variable exponents that depend on the unknown solution and its gradient. Using these embeddings and an abstract critical point theorem, we prove the existence and multiplicity of weak solutions for such problems associated with this new operator in the whole space $\mathbb{R}^d$. This work can be seen as a continuation of the recent paper by Bahrouni--Bahrouni--Missaoui--Rădulescu \cite{Bahrouni-Bahrouni-Missaoui-Radulescu-2024}.

math.AP↗