Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908579608395776 |
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| author | Ó, J. M. Do Freire, R. F. Giacomoni, J. Medeiros, E. S. |
| author_facet | Ó, J. M. Do Freire, R. F. Giacomoni, J. Medeiros, E. S. |
| contents | In this paper we study the problem $-\mathrm{div}(ρ(x_N)\nabla u)=a|u|^{p-2}u$ in $\mathbb{R}^N_+$, $-\partial u/\partial x_N=b|u|^{q-2}u$ in $\mathbb{R}^{N-1}$ where $a,b \in \mathbb{R}$, $p,q\in (1,\infty)$ and $ρ$ is a positive weight. We establish regularity results for weak solutions and, using a variational approach combined with a new Pohozaev-type identity, we show that the introduction of the weighted operator $-\mathrm{div}(ρ(x_N)\nabla u)$ can reverse the known solvability behavior of the classical Laplacian case. Specifically, we identify regimes where the problem admits solutions despite nonexistence for the corresponding case with $-Δ$, and vice versa, thus inverting the classical existence and nonexistence results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space Ó, J. M. Do Freire, R. F. Giacomoni, J. Medeiros, E. S. Analysis of PDEs 35J55, 35J50, 37K05 In this paper we study the problem $-\mathrm{div}(ρ(x_N)\nabla u)=a|u|^{p-2}u$ in $\mathbb{R}^N_+$, $-\partial u/\partial x_N=b|u|^{q-2}u$ in $\mathbb{R}^{N-1}$ where $a,b \in \mathbb{R}$, $p,q\in (1,\infty)$ and $ρ$ is a positive weight. We establish regularity results for weak solutions and, using a variational approach combined with a new Pohozaev-type identity, we show that the introduction of the weighted operator $-\mathrm{div}(ρ(x_N)\nabla u)$ can reverse the known solvability behavior of the classical Laplacian case. Specifically, we identify regimes where the problem admits solutions despite nonexistence for the corresponding case with $-Δ$, and vice versa, thus inverting the classical existence and nonexistence results. |
| title | Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space |
| topic | Analysis of PDEs 35J55, 35J50, 37K05 |
| url | https://arxiv.org/abs/2510.05999 |