Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911027948421120 |
|---|---|
| author | Chen, Lu Jiang, Rou Lu, Guozhen Zhu, Maochun |
| author_facet | Chen, Lu Jiang, Rou Lu, Guozhen Zhu, Maochun |
| contents | In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:\[ S_Ω(λ,p)=\sup_{u\in H_{0}^{1}(Ω),\Vert\nabla u\Vert _{L^{2}\left( Ω\right) }\leq 1}\int_Ω\left( e^{4πu^{2}}-λ|u|^{p}\right) dx, \] where $1\leq p<\infty$ and $Ω$ is a bounded domain in $\mathbb{R}^2$. Our results demonstrate that there exists a threshold $λ^{\ast}(p)>0$ such that $S_Ω(λ,p)$ is attainable if $λ<λ^{\ast}(p)$, but unattainable if $λ>λ^{\ast}(p)$ when $p\in[1,2]$. For $p>2$, however, we show that $S_Ω(λ,p)$ is always attainable for any $λ\in \mathbb{R}$. These results are achieved through a refined blow-up analysis, which allow us to establish a sharp Dirichlet energy expansion formula for sequences of solutions to the corresponding Euler-Lagrange equations. The asymmetric nature of our problem poses significant challenges to our analysis. To address these, we will establish an appropriate comparison principle between radial and non-radial solutions of the associated Euler-Lagrange equations. Our study establishes a complete characterization of how $L^p$-type perturbations influence the existence of extremals for critical Trudinger-Moser inequalities on any bounded planar domains, this extends the classical Brezis-Nirenberg problem framework to the two-dimensional settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains Chen, Lu Jiang, Rou Lu, Guozhen Zhu, Maochun Analysis of PDEs 35B44, 35B40, 35J61, 35B33 In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:\[ S_Ω(λ,p)=\sup_{u\in H_{0}^{1}(Ω),\Vert\nabla u\Vert _{L^{2}\left( Ω\right) }\leq 1}\int_Ω\left( e^{4πu^{2}}-λ|u|^{p}\right) dx, \] where $1\leq p<\infty$ and $Ω$ is a bounded domain in $\mathbb{R}^2$. Our results demonstrate that there exists a threshold $λ^{\ast}(p)>0$ such that $S_Ω(λ,p)$ is attainable if $λ<λ^{\ast}(p)$, but unattainable if $λ>λ^{\ast}(p)$ when $p\in[1,2]$. For $p>2$, however, we show that $S_Ω(λ,p)$ is always attainable for any $λ\in \mathbb{R}$. These results are achieved through a refined blow-up analysis, which allow us to establish a sharp Dirichlet energy expansion formula for sequences of solutions to the corresponding Euler-Lagrange equations. The asymmetric nature of our problem poses significant challenges to our analysis. To address these, we will establish an appropriate comparison principle between radial and non-radial solutions of the associated Euler-Lagrange equations. Our study establishes a complete characterization of how $L^p$-type perturbations influence the existence of extremals for critical Trudinger-Moser inequalities on any bounded planar domains, this extends the classical Brezis-Nirenberg problem framework to the two-dimensional settings. |
| title | Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains |
| topic | Analysis of PDEs 35B44, 35B40, 35J61, 35B33 |
| url | https://arxiv.org/abs/2506.23076 |