E. Azroul | Searcharxiv

arXiv2023

Eigenvalue type problem in $s(.,.)$-fractional Musielak-Sobolev spaces

In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(\Omega)$. Next, by means of Ekeland's variational principal, we show that there exists $\lambda_*>0$ such that any $\lambda\in(0, \lambda_*)$ is an eigenvalue for the following problem $$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left( -\Delta\right)^{s(x,.)}_{a_{(x,.)}} u = \lambda |u|^{q(x)-2}u &\quad {\rm in}\ \Omega, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus \Omega, \end{array} \right. $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary.

math.AP↗

arXiv2020

Quasi-linear elliptic equations with data in $L^{1}$ on a compact Riemannian manifold

This work is dedicated to the study of quasi-linear elliptic problems with $L^1$ data, the simple model will be the next equation on $ (M,g) $ a compact Riemannian manifold. $$-\Delta_{p} u=f$$ Where $f\in L^{1}(M) $ .Our goal is to develop the functional framework and tools that are necessary to prove the existence and the uniqueness of the solution for the previous problem. Notice that our argument can be used to deal with a more general class of quasi-linear equations.

math.AP↗