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| Format: | Preprint |
| Veröffentlicht: |
2021
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| Online-Zugang: | https://arxiv.org/abs/2111.13417 |
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| _version_ | 1866908344104517632 |
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| author | De Nitti, Nicola König, Tobias |
| author_facet | De Nitti, Nicola König, Tobias |
| contents | For $s \in (0,1)$ and a bounded open set $Ω\subset \mathbb R^N$ with $N > 2s$, we study the fractional Brezis--Nirenberg type minimization problem of finding $$ S(a) := \inf \frac{\int_{\mathbb R^N} |(-Δ)^{s/2} u|^2 + \int_Ωa u^2}{\left( \int_Ωu^\frac{2N}{N-2s} \right)^\frac{N-2s}{N}}, $$ where the infimum is taken over all functions $u \in H^s(\mathbb R^N)$ that vanish outside $Ω$. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions $N \in (2s, 4s)$, we prove that the Robin function $ϕ_a$ satisfies $\inf_{x \in Ω} ϕ_a(x) = 0$, which extends a result obtained by Druet for $s = 1$. In dimensions $N \in (8s/3, 4s)$, we then study the asymptotics of the fractional Brezis--Nirenberg energy $S(a + \varepsilon V)$ for some $V \in L^\infty(Ω)$ as $\varepsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_13417 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension De Nitti, Nicola König, Tobias Analysis of PDEs For $s \in (0,1)$ and a bounded open set $Ω\subset \mathbb R^N$ with $N > 2s$, we study the fractional Brezis--Nirenberg type minimization problem of finding $$ S(a) := \inf \frac{\int_{\mathbb R^N} |(-Δ)^{s/2} u|^2 + \int_Ωa u^2}{\left( \int_Ωu^\frac{2N}{N-2s} \right)^\frac{N-2s}{N}}, $$ where the infimum is taken over all functions $u \in H^s(\mathbb R^N)$ that vanish outside $Ω$. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions $N \in (2s, 4s)$, we prove that the Robin function $ϕ_a$ satisfies $\inf_{x \in Ω} ϕ_a(x) = 0$, which extends a result obtained by Druet for $s = 1$. In dimensions $N \in (8s/3, 4s)$, we then study the asymptotics of the fractional Brezis--Nirenberg energy $S(a + \varepsilon V)$ for some $V \in L^\infty(Ω)$ as $\varepsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points. |
| title | Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2111.13417 |