The Hénon equation in Orlicz-Sobolev spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914051075866624 |
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| author | Ochoa, Pablo Salort, Ariel |
| author_facet | Ochoa, Pablo Salort, Ariel |
| contents | In this paper, we consider the Hénon problem in the setting of Orlicz-Sobolev spaces: \begin{equation*} \begin{cases} -Δ_g u= |x|^αh( u) \quad \text{in }B\\ u>0 \quad \text{in }B\\ u= 0 \quad \text{on }\partial B\\ \end{cases} \end{equation*}where $B$ is the unit ball in $\mathbb{R}^n$, $g=G'$, $h=H'$ are N-functions and the operator $-Δ_g$ is the $g$-Laplacian. We show that the symmetric term $|x|^α$, for $α>0$, allows to have radial solutions even for supercritical $H$, generalizing results for the classical Hénon equation. We also show that radial solutions are indeed bounded. Finally, we state a Pohozaev's identity in Orlicz-Sobolev spaces that we apply to get a range in $α$ for which the problem has no bounded solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hénon equation in Orlicz-Sobolev spaces Ochoa, Pablo Salort, Ariel Analysis of PDEs In this paper, we consider the Hénon problem in the setting of Orlicz-Sobolev spaces: \begin{equation*} \begin{cases} -Δ_g u= |x|^αh( u) \quad \text{in }B\\ u>0 \quad \text{in }B\\ u= 0 \quad \text{on }\partial B\\ \end{cases} \end{equation*}where $B$ is the unit ball in $\mathbb{R}^n$, $g=G'$, $h=H'$ are N-functions and the operator $-Δ_g$ is the $g$-Laplacian. We show that the symmetric term $|x|^α$, for $α>0$, allows to have radial solutions even for supercritical $H$, generalizing results for the classical Hénon equation. We also show that radial solutions are indeed bounded. Finally, we state a Pohozaev's identity in Orlicz-Sobolev spaces that we apply to get a range in $α$ for which the problem has no bounded solutions. |
| title | The Hénon equation in Orlicz-Sobolev spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.17923 |