Past and recent contributions to indefinite sublinear elliptic problems
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866911760457400320 |
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| author | Kaufmann, Uriel Quoirin, Humberto Ramos Umezu, Kenichiro |
| author_facet | Kaufmann, Uriel Quoirin, Humberto Ramos Umezu, Kenichiro |
| contents | We review the indefinite sublinear elliptic equation $-Δu=a(x)u^{q}$ in a smooth bounded domain $Ω\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_01284 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Past and recent contributions to indefinite sublinear elliptic problems Kaufmann, Uriel Quoirin, Humberto Ramos Umezu, Kenichiro Analysis of PDEs We review the indefinite sublinear elliptic equation $-Δu=a(x)u^{q}$ in a smooth bounded domain $Ω\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched. |
| title | Past and recent contributions to indefinite sublinear elliptic problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2004.01284 |