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Hichem Ounaies

Publications and source records attributed to Hichem Ounaies.

13 recordsLinked to original sources
arXiv2026

A Class of De Giorgi Type and Hölder Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces

In this paper, we introduce a new class of De Giorgi type functions, denoted by \(\mathcal{B}_{G(x,t)}\), and establish the Hölder continuity of its elements under suitable additional assumptions on the generalized \textnormal{N}-function \(G(x,t)\). As an application, we prove the Hölder continuity of solutions to quasilinear equations whose principal part is in divergence form with \(G(x,t)\)-growth conditions, including both critical and standard growth cases. The novelty of our work lies in the generalization of the Hölder continuity results previously known for variable exponent \cite[X, Fan and D. Zhao]{Fan1999} and Orlicz \cite[G. M. Lieberman]{Li1991} problems. Moreover, our results encompass a wide variety of quasilinear equations.

math.AP↗
arXiv2025

Orlicz-Sobolev versus Hölder local minimizer for nonlinear Robin problems

In this paper, we establish a regularity results for weak solutions of Robin problems driven by the well-known Orlicz $g$-Laplacian operator. Precisely, by using a suitable variation of the Moser iteration technique, we prove that every weak solution of our problem is bounded. Moreover, we combine this result with the Lieberman regularity theorem, to show that every $C^1(\overlineΩ)$-local minimizer is also a $W^{1,G}(Ω)$-local minimizer for the corresponding energy functional of Robin-Orlicz problem.

math.AP↗
arXiv2023

Espaces d'Orlicz, Orlicz-Sobolev et application aux E-D-P

In this article, we will define the Orlicz space and the Orlicz-Sobolev space, and develop their topological properties. We will also examine their applications to partial differential equations (PDEs), with an emphasis on the use of certain variational methods.

math.FA↗
arXiv2023

On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in $\mathbb{R}^d$. As an application, using the variational methods, we obtain the existence of nontrivial weak solution for the following Schrödinger equation $$ (-Δ)_{g_{x,y}}^s u+V(x)g(x,x,u)=b(x)\vert u\vert^{p(x)-2}u,\ \text{for all}\ x\in \mathbb{R}^d,$$ where $(-Δ)_{g_{x,y}}^s$ is the fractional Museilak $g_{x,y}$-Laplacian, $V$ is a potential function, $b\in L^{δ^{'}(x)}(\mathbb{R}^d)$, and $p,δ\in C\left(\mathbb{R}^d,(1,+\infty)\right)\cap L^{\infty}(\mathbb{R}^d)$. We would like to mention that the theory of the fractional Musielak-Sobolev spaces is in a developing state and there are few papers in this topic, see \cite{M1,M8,M9}. Note that, all these latter works dealt with bounded case and there are no results devoted for the fractional Musielak-Sobolev spaces in $\mathbb{R}^d$. Since the embedding results are crucial in applying variational methods, this work will provide a bridge between the fractional Mueislak-Sobolev theory and PDE's.

math.AP↗
arXiv2022

Problems involving the fractional $g$-Laplacian with Lack of Compactness

In this paper we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space $W^{s, G}\left(\mathbb{R}^{N}\right)$ consisting of radial functions, our target embedding spaces are of Orlicz type. Also, we prove a Lions and Lieb type results for $W^{s,G}\left(\mathbb{R}^{N}\right)$ that works together in a particular way to get a sequence whose the weak limit is nontrivial. As an application, we study the existence of solutions to Quasilinear elliptic problems in the whole space $\mathbb{R}^N$ involving the fractional $g-$Laplacian operator, where the conjugated function $\widetilde{G}$ of $G$ doesn't satisfy the $Δ_2$-condition.

math.AP↗
arXiv2022

Ground state and nodal solutions for fractional Orlicz problems with lack of regularity and without the Ambrosetti-Rabinowitz condition

We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows \begin{equation}\label{PP} (-\triangle_{g})^αu+g(u)=K(x)f(x,u),\ \ \text{in}\ \mathbb{R}^{d},\tag{P} \end{equation} where $d\geq 3,\ (-\triangle_{g})^α$ is the fractional Orlicz g-Laplace operator, $f:\mathbb{R}^d\times\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $K$ is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term $f$, we prove that the problem \eqref{PP} has a ground state of fixed sign and a nodal (or sign-changing) solutions.

math.AP↗
arXiv2021

Least-energy nodal solutions of nonlinear equations with fractional Orlicz-Sobolev spaces

In our work, we prove the existence of least-energy nodal solutions for nonlinear equations in which the new fractional Orlicz Laplacian is present. Precisely, we prove a compact embeddings result for weighted fractional Orlicz-Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show our desired result.

math.AP↗
arXiv2020

Variational Eigenvalues of the fractional $g$-Laplacian

In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional $g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks must be overcome, leading to some results that contrast with the case of power functions.

math.AP↗
arXiv2019

Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}u+V(x)m(u)=f(x,u),\ x\in\mathbb{R}^{d},$$ where $0<s<1$, $d\geq2$ and $(-\triangle)^{s}_{m}$ is the fractional $M$-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.

math.AP↗
arXiv2019

Basic results of fractional Orlicz-Sobolev space and applications to non-local problems

In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional $M-$Laplacian operator.

math.AP↗
arXiv2019

Infinitely many solutions for a class of fractional Orlicz-Sobolev Schrödinger equations

In the present paper, we deal with a new compact embedding theorem for a subspace of the new fractional Orlicz-Sobolev spaces. We also establish some useful inequalities which yields to apply the variational methods. Using these abstract results, we study the existence of infinitely many nontrivial solutions for a class of fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}+V(x)m(u)u=f(x,u),\ x\in\mathbb{R}^{N},$$ where $s\in ]0,1[$, $N\geq2$, $(-\triangle)^{s}_{m}$ is fractional $M$-Laplace operator and the nonlinearity $f$ is sublinear as $|u| \rightarrow\infty$. The proof is based on the variant Fountain theorem established by Zou.

math.AP↗
arXiv2015

Lyapunov-Sylvester Operators for Numerical Solutions of NLS Equation

In the present paper a numerical method is developed to approximate the solution of two-dimensional NLS equation in the presence of a singular potential. The method leads to Lyapunov-Syslvester algebraic operators that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and stable using the based on Lyapunov criterion, lax equivalence theorem and the properties of the Lyapunov-Syslvester operators.

math.NA↗