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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.17499 |
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| _version_ | 1866915944653127680 |
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| author | Li, Jiamo Lu, Qikai Yang, Minbo |
| author_facet | Li, Jiamo Lu, Qikai Yang, Minbo |
| contents | For dimensions $n\geq8$, we are concerned with the quotient functional of the biharmonic Brézis-Nirenberg problem under the Navier boundary condition
$$
S(\varepsilon V):=\inf_{0\not\equiv u\in H^2(Ω)\cap H_0^1(Ω)}\frac{\int_Ω|Δu|^2dx+\varepsilon\int_ΩV|u|^2dx}{\big(\int_Ω|u|^{2^\star}dx\big)^{2/2^\star}},
$$
where $2^\star=\frac{2n}{n-4}$ is the critical Sobolev exponent of the embedding $H^2(Ω)\cap H_0^1(Ω)\hookrightarrow L^{2^\star}(Ω)$, $Ω\subset\mathbb{R}^n$ is a bounded open set and $V:\overlineΩ\rightarrow\mathbb{R}$ is a continuous function. Under certain assumptions on $V$, we establish sharp asymptotics for the energy difference $S(0)-S(\varepsilon V)$, as $\varepsilon\rightarrow0^+$, by means of matching upper and lower bound estimates. Moreover, we give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the blow-up rate and the location of concentration points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17499 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Energy asymptotics and blow-up phenomena for biharmonic Brézis-Nirenberg problem Li, Jiamo Lu, Qikai Yang, Minbo Analysis of PDEs 35J40, 35A15, 35B44 For dimensions $n\geq8$, we are concerned with the quotient functional of the biharmonic Brézis-Nirenberg problem under the Navier boundary condition $$ S(\varepsilon V):=\inf_{0\not\equiv u\in H^2(Ω)\cap H_0^1(Ω)}\frac{\int_Ω|Δu|^2dx+\varepsilon\int_ΩV|u|^2dx}{\big(\int_Ω|u|^{2^\star}dx\big)^{2/2^\star}}, $$ where $2^\star=\frac{2n}{n-4}$ is the critical Sobolev exponent of the embedding $H^2(Ω)\cap H_0^1(Ω)\hookrightarrow L^{2^\star}(Ω)$, $Ω\subset\mathbb{R}^n$ is a bounded open set and $V:\overlineΩ\rightarrow\mathbb{R}$ is a continuous function. Under certain assumptions on $V$, we establish sharp asymptotics for the energy difference $S(0)-S(\varepsilon V)$, as $\varepsilon\rightarrow0^+$, by means of matching upper and lower bound estimates. Moreover, we give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the blow-up rate and the location of concentration points. |
| title | Energy asymptotics and blow-up phenomena for biharmonic Brézis-Nirenberg problem |
| topic | Analysis of PDEs 35J40, 35A15, 35B44 |
| url | https://arxiv.org/abs/2604.17499 |