arXiv subjects

A. Aberqi

Publications and source records attributed to A. Aberqi.

6 recordsLinked to original sources

arXiv2023

Fractional Sobolev spaces with kernel function on compact Riemannian manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional $p(\cdot, \cdot)$-Laplacian operator. As one of the main tools, topological degree theory is applied.

math.AP↗

arXiv2023

Singular fractional double-phase problems with variable exponent via Morse's theory

In this manuscript, we deal with a class of fractional non-local problems involving a singular term and vanishing potential of the form: \begin{eqnarray*} \begin{gathered} \left\{\begin{array}{llll} \mathcal{L}^{s_{1}, s_{2}}_{p(\mathrm{x}, .), q(\mathrm{x}, .)}\mathrm{w}(\mathrm{x})&= \displaystyle\frac{g(\mathrm{x}, \mathrm{w}(\mathrm{x}))}{ \mathrm{w}(\mathrm{x})^{ξ(\mathrm{x})}} + \mathcal{V}(\mathrm{x}) \vert \mathrm{w}(\mathrm{x}) \vert^{σ(\mathrm{x})-2} \mathrm{w}(\mathrm{x}) & \text { in } & \mathcal{U}, \\ \hspace{2cm} \mathrm{w}&> 0 & \text { in }& \mathcal{U},\\ \hspace{2cm} \mathrm{w}&=0 & \text { in }& \mathbb{R}^{N} \backslash \mathcal{U}, \end{array}\right. \end{gathered} \end{eqnarray*} where, $ \mathcal{L}^{s_{1}, s_{2}}_{p(\mathrm{x}, .), q(\mathrm{x}, .)}$ is a $\left(p(\mathrm{x}, .), q(\mathrm{x}, .)\right)$-fractional double-phase operator with $ s_{1},s_{2 }\in \left( 0, 1\right)$, $g,$ and $\mathcal{V}$ are functions that satisfy some conditions. The strategy of the proof for these results is to approach the problem proximatively and calculate the critical groups. Moreover, using Morse's theory to prove our problem has infinitely many solutions.

math.AP↗

arXiv2022

Frational p-Laplacian on Compact Riemannian Manifold

In this paper, we investigate the existence and uniqueness of a non-trivial solution for a class of nonlocal equations involving the fractional $p$-Laplacian operator defined on compact Riemannian manifold, namely, \begin{eqnarray}\label{k1} \begin{gathered} \left\{\begin{array}{lll} (-Δ_g)^s_p u(x)+ \left| u \right|^{p-2} u= f(x,u) & \text { in }& Ω, \hspace{3,4cm} u=0 & \text{in }& M\setminusΩ, \end{array}\right. \end{gathered} \end{eqnarray} and $Ω$ is an open bounded subset of M with a smooth boundary.

math.AP↗

arXiv2022

Weak solvability of nonlinear elliptic equations involving variable exponents

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the $( p( m ), \, q( m ) )-$ equation and the nonlinearity is superlinear but does not fulfil the Ambrosetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a $( p( m ), \, q( m ) )$ equation that highlights the applicability of our theoretical results is also provided.

math.DG↗

arXiv2021

Descret Solution for a Nonlinear Parabolic Equations with Diffusion Terms in Museilak-Spaces

In this paper, we study a class of nonlinear evolution equations with damping arising in fluid dynamics and rheology. The nonlinear term is monotone and possesses a convex potential but exhibits non-standard growth. The appropriate functional framework for such equations is the modular Museilak-Spaces. We prove the existence and uniqueness of a weak solution using an approximation approach by combining an internal approximation with the backward Euler scheme, also a priori error estimate for the temporal semi-discretization is given.

math.AP↗

arXiv2020

On some $\overrightarrow{p(x)}$ anisotropic elliptic equations in unbounded domain

We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $Ω\subset \>I\!\!R^{N}\>(N \geq 2)\>$. We prove the existence of entropy solutions avoiding sign condition and coercivity on the lowers order terms.

math.AP↗