Positive normalized solutions to a singular elliptic equation with a $L^2$-supercritical nonlinearity
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912854197665792 |
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| author | Chen, Siyu Chang, Xiaojun Zhou, Jiazheng |
| author_facet | Chen, Siyu Chang, Xiaojun Zhou, Jiazheng |
| contents | This paper studies the existence of positive normalized solutions to the singular elliptic equation \[ -Δu + λu = u^{-r} + u^{p-1} \quad \text{in } Ω, \] with the Dirichlet boundary condition $u=0$ on $\partialΩ$ and the normalization constraint $\int_Ωu^2\,dx = ρ$. Here $Ω\subset\mathbb{R}^N$ ($N\ge3$) is a smooth bounded domain, $0<r<1$, $2+\frac{4}{N}<p<2^*$, where $2^*$ is the critical Sobolev exponent, and $λ\in\mathbb{R}$ is a Lagrange multiplier. We obtain that for sufficiently small $ρ>0$, the problem admits a positive solution $(λ,u)\in\mathbb{R}\times H_0^1(Ω)$. The proof is based on a variational approach using a regularized functional and a careful analysis of the limiting process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20200 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive normalized solutions to a singular elliptic equation with a $L^2$-supercritical nonlinearity Chen, Siyu Chang, Xiaojun Zhou, Jiazheng Analysis of PDEs 35B09, 35J20, 35J60, 35J75 This paper studies the existence of positive normalized solutions to the singular elliptic equation \[ -Δu + λu = u^{-r} + u^{p-1} \quad \text{in } Ω, \] with the Dirichlet boundary condition $u=0$ on $\partialΩ$ and the normalization constraint $\int_Ωu^2\,dx = ρ$. Here $Ω\subset\mathbb{R}^N$ ($N\ge3$) is a smooth bounded domain, $0<r<1$, $2+\frac{4}{N}<p<2^*$, where $2^*$ is the critical Sobolev exponent, and $λ\in\mathbb{R}$ is a Lagrange multiplier. We obtain that for sufficiently small $ρ>0$, the problem admits a positive solution $(λ,u)\in\mathbb{R}\times H_0^1(Ω)$. The proof is based on a variational approach using a regularized functional and a careful analysis of the limiting process. |
| title | Positive normalized solutions to a singular elliptic equation with a $L^2$-supercritical nonlinearity |
| topic | Analysis of PDEs 35B09, 35J20, 35J60, 35J75 |
| url | https://arxiv.org/abs/2601.20200 |