Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition
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| Format: | Preprint |
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2024
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| _version_ | 1866929230374240256 |
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| author | Bonder, Julian Fernandez Spedaletti, Juan F. |
| author_facet | Bonder, Julian Fernandez Spedaletti, Juan F. |
| contents | In this work we analyze the eigenvalue problem associated to the fractional $m-$Laplacian, defined as $$ (-Δ_m)^s u(x):=2\text{p.v.}\int_{{\mathbb R}^n} m\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{(u(x)-u(y))}{|u(x)-u(y)|}\frac{dy}{|x-y|^{n+s}}, $$ This operator serves as a model for nonlocal, nonstandard growth diffusion problems. In contrast to previous analyses, we explore the eigenvalue problem without presuming the $Δ_2$ condition on $M$ -- the primitive function of $m$. Our results show the existence of a sequence of eigenvalues $λ_k\to\infty$. This research contributes to advancing our understanding of nonlocal diffusion models, specifically those characterized by the fractional $m-$Laplacian, by relaxing the constraints imposed by the $Δ_2$ condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_18041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition Bonder, Julian Fernandez Spedaletti, Juan F. Analysis of PDEs 35J62, 35P30, 46E30 In this work we analyze the eigenvalue problem associated to the fractional $m-$Laplacian, defined as $$ (-Δ_m)^s u(x):=2\text{p.v.}\int_{{\mathbb R}^n} m\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{(u(x)-u(y))}{|u(x)-u(y)|}\frac{dy}{|x-y|^{n+s}}, $$ This operator serves as a model for nonlocal, nonstandard growth diffusion problems. In contrast to previous analyses, we explore the eigenvalue problem without presuming the $Δ_2$ condition on $M$ -- the primitive function of $m$. Our results show the existence of a sequence of eigenvalues $λ_k\to\infty$. This research contributes to advancing our understanding of nonlocal diffusion models, specifically those characterized by the fractional $m-$Laplacian, by relaxing the constraints imposed by the $Δ_2$ condition. |
| title | Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition |
| topic | Analysis of PDEs 35J62, 35P30, 46E30 |
| url | https://arxiv.org/abs/2401.18041 |