arXiv subjects

Mohammed Srati

Publications and source records attributed to Mohammed Srati.

8 recordsLinked to original sources

arXiv2025

On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems

In this paper, we introduce and study a new class of fractional modular function spaces, called \emph{Fractional Anisotropic Musielak--Sobolev Spaces}, which generalize both the fractional Anisotropic Orlicz--Sobolev spaces and the Anisotropic fractional Sobolev spaces with variable exponent. These spaces are designed to handle anisotropic and heterogeneous behaviors that naturally arise in nonlocal and nonlinear models. We develop their fundamental properties and embedding results, establishing a solid variational framework. As an application, we investigate a class of nonlocal anisotropic eigenvalue problems involving variable growth and direction-dependent fractional integro-differential operators. We prove the existence of eigenvalues by means of critical point theory and modular analysis. Our results extend and unify several existing models in the theory of nonlocal partial differential equations.

math.AP↗

arXiv2024

Nonlocal double phase Neumann and Robin problem with variable $s(\cdot,\cdot)-$order

In this paper, we develop some properties of the $a_{x,y}(\cdot)$-Neumann derivative for the nonlocal $s(\cdot,\cdot)$-order operator in fractional Musielak-Sobolev spaces with variable $s(\cdot,\cdot)-$order. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions to the following double phase Neumann and Robin problem with variable $s(\cdot,\cdot)-$order: $$\left\{\begin{array} (-Δ)^{s_1(x,\cdot)}_{a^1_{(x,\cdot)}} u+(-Δ)^{s_2(x,\cdot)}_{a^2_{(x,\cdot)}} u +\widehat{a}^1_x(|u|)u+\widehat{a}^2_x(|u|)u &= λf(x,u) \quad {\rm in\ } Ω, \\ \mathcal{N}^{s_1(x,\cdot)}_{a^1(x,\cdot)}u+\mathcal{N}^{s_2(x,\cdot)}_{a^2(x,\cdot)}u+β(x)\left( \widehat{a}^1_x(|u|)u+\widehat{a}^2_x(|u|)u \right) &= 0 \quad {\rm in\ } \mathbb{R}^N\setminus Ω, \end{array} \right. $$ where $(-Δ)^{s_i(x,\cdot)}_{a^i_{(x,\cdot)}}$ and $\mathcal{N}^{s_i(x,\cdot)}_{a^i(x,\cdot)}$ denote the variable $s_i(\cdot,\cdot)$-order fractional Laplace operator and the nonlocal normal $a_i(\cdot,\cdot)$-derivative of $s_i(\cdot,\cdot)$-order, respectively.

math.AP↗

arXiv2023

Eigenvalue problems in fractional anisotropic Orlicz-Sobolev spaces

In this paper, we introduce the fractional anisotropic Orlicz-Sobolev spaces, and by using some variational methods, we establish the existence or non-existence of eigenvalues of fractional anisotropic problems involving a nonlocal integro-differential operator of elliptic type. In each case, the competition between the growth rates of the anisotropic coefficients plays an essential role in the description of the set of eigenvalues.

math.AP↗

arXiv2022

Nonlocal problems with Neumann and Robin boundary condition in fractional Musielak-Sobolev spaces

In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.

math.AP↗

arXiv2020

Embedding and extension results in Fractional Musielak-Sobolev spaces

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces $W^sL_{\varPhi_{x,y}}$ such that the generalized Poincaré type inequality and some continuous and compact embedding theorems of these spaces. Moreover, we prove that any function in $W^sL_{\varPhi_{x,y}}(Ω)$ may be extended to a function in $W^sL_{\varPhi_{x,y}}(\R^N)$, with $Ω\subset \R^N$ is a bounded domain of class $C^{0,1}$. In addition, we establish a result relates to the complemented subspace in $W^s{L_{\varPhi_{x,y}}}\left( \R^N\right)$. As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data.

math.AP↗

arXiv2019

Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces $W^{s,p}$ to include the general case $W^sL_A$, where $A$ is an N-function and $s\in (0,1)$. We are concerned with some qualitative properties of the space $W^sL_A$ (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.

math.AP↗

arXiv2019

Existence of solutions for a nonlocal Kirchhoff type problem in Fractional Orlicz-Sobolev spaces

In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$ (D_{K,A}) \hspace*{0.5cm} \left\{ \begin{array}{clclc} M\left( \displaystyle \int_{\R^{2N}}A\left( [u(x)-u(y)] K(x,y)\right) dxdy\right) \mathcal{L}^K_A u & = & f(x,u) & \text{ in }& Ω, \hspace*{7cm} u & = & 0 \hspace*{0.2cm} \hspace*{0.2cm} & \text{ in } & \R^N\setminus Ω. \label{eq1} \end{array} \right. $$ } Where $\mathcal{L}^K_A$ is a nonlocal operator with singular kernel $K$ and $A$ is an $N$-function, $Ω$ is an open bounded subset in $\R^N$ with Lipschitz boundary $\partial Ω$.

math.AP↗

arXiv2018

Introduction to fractional Orlicz-Sobolev spaces

In this paper, we define the fractional Orlicz-Sobolev spaces, and we prove some important results of these spaces. The main result is to show the continuous and compact embedding for these spaces. As an application, we prove the existence and uniqueness of a solution for a non local problem involving the fractional M-Laplacian operator.

math.AP↗