On the nodal set of solutions to some sublinear equations without homogeneity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911810952626176 |
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| author | Soave, Nicola Tortone, Giorgio |
| author_facet | Soave, Nicola Tortone, Giorgio |
| contents | We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -Δu = λ_+(u^+)^{p-1}-λ_-(u^-)^{q-1} \] where $1 \le p<q<2$, $λ_+ >0$, $λ_- \ge 0$. The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_10161 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the nodal set of solutions to some sublinear equations without homogeneity Soave, Nicola Tortone, Giorgio Analysis of PDEs We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -Δu = λ_+(u^+)^{p-1}-λ_-(u^-)^{q-1} \] where $1 \le p<q<2$, $λ_+ >0$, $λ_- \ge 0$. The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions. |
| title | On the nodal set of solutions to some sublinear equations without homogeneity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2204.10161 |