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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2301.00467 |
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| _version_ | 1866929237321056256 |
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| author | Azroul, E. Benkirane, A. Srati, M. |
| author_facet | Azroul, E. Benkirane, A. Srati, M. |
| contents | In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(Ω)$. Next, by means of Ekeland's variational principal, we show that there exists $λ_*>0$ such that any $λ\in(0, λ_*)$ is an eigenvalue for the following problem $$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left( -Δ\right)^{s(x,.)}_{a_{(x,.)}} u = λ|u|^{q(x)-2}u &\quad {\rm in}\ Ω, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus Ω, \end{array} \right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00467 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eigenvalue type problem in $s(.,.)$-fractional Musielak-Sobolev spaces Azroul, E. Benkirane, A. Srati, M. Analysis of PDEs In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(Ω)$. Next, by means of Ekeland's variational principal, we show that there exists $λ_*>0$ such that any $λ\in(0, λ_*)$ is an eigenvalue for the following problem $$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left( -Δ\right)^{s(x,.)}_{a_{(x,.)}} u = λ|u|^{q(x)-2}u &\quad {\rm in}\ Ω, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus Ω, \end{array} \right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary. |
| title | Eigenvalue type problem in $s(.,.)$-fractional Musielak-Sobolev spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2301.00467 |