Singular semilinear elliptic equations in nondivergence form
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913090773188608 |
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| author | Kałamajska, Agnieszka Peša, Dalimil Rutkowski, Artur |
| author_facet | Kałamajska, Agnieszka Peša, Dalimil Rutkowski, Artur |
| contents | We study the singular semilinear equation $-Pu = \frac{f}{u^γ}$ on a bounded domain $Ω$ with Dirichlet condition $u \equiv 0$ on $\partial Ω$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $Ω\in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(Ω)$, under the assumptions that $Ω\in C^2$ and $P$ has $C^2$ coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo--Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03704 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Singular semilinear elliptic equations in nondivergence form Kałamajska, Agnieszka Peša, Dalimil Rutkowski, Artur Analysis of PDEs Functional Analysis Primary 35J75, 35J61, Secondary 35D30, 35J08 We study the singular semilinear equation $-Pu = \frac{f}{u^γ}$ on a bounded domain $Ω$ with Dirichlet condition $u \equiv 0$ on $\partial Ω$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $Ω\in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(Ω)$, under the assumptions that $Ω\in C^2$ and $P$ has $C^2$ coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo--Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities. |
| title | Singular semilinear elliptic equations in nondivergence form |
| topic | Analysis of PDEs Functional Analysis Primary 35J75, 35J61, Secondary 35D30, 35J08 |
| url | https://arxiv.org/abs/2605.03704 |