Nonlinear eigenvalue problems for a biharmonic operator in Orlicz-Sobolev spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916465962123264 |
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| author | Ochoa, Pablo Silva, Analía |
| author_facet | Ochoa, Pablo Silva, Analía |
| contents | In this paper, we introduce a new higher-order Laplacian operator in the framework of Orlicz-Sobolev spaces, the biharmonic g-Laplacian $$Δ_g^2 u:=Δ\left(\dfrac{g(|Δu|)}{|Δu|} Δu\right),$$ where $g=G'$, with $G$ an N-function. This operator is a generalization of the so called bi-harmonic Laplacian $Δ^2$. Here, we also established basic functional properties of $Δ_g^2$, which can be applied to existence results. Afterwards, we study the eigenvalues of $Δ_g^2$, which depend on normalisation conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to $Δ_g^2$ and we show regimes where the corresponding spectrum concentrate at $0$, $\infty$ or coincide with $(0, \infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01276 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear eigenvalue problems for a biharmonic operator in Orlicz-Sobolev spaces Ochoa, Pablo Silva, Analía Analysis of PDEs 46E30, 35P30, 35D30 In this paper, we introduce a new higher-order Laplacian operator in the framework of Orlicz-Sobolev spaces, the biharmonic g-Laplacian $$Δ_g^2 u:=Δ\left(\dfrac{g(|Δu|)}{|Δu|} Δu\right),$$ where $g=G'$, with $G$ an N-function. This operator is a generalization of the so called bi-harmonic Laplacian $Δ^2$. Here, we also established basic functional properties of $Δ_g^2$, which can be applied to existence results. Afterwards, we study the eigenvalues of $Δ_g^2$, which depend on normalisation conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to $Δ_g^2$ and we show regimes where the corresponding spectrum concentrate at $0$, $\infty$ or coincide with $(0, \infty)$. |
| title | Nonlinear eigenvalue problems for a biharmonic operator in Orlicz-Sobolev spaces |
| topic | Analysis of PDEs 46E30, 35P30, 35D30 |
| url | https://arxiv.org/abs/2411.01276 |