Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2502.16505 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917934355447808 |
|---|---|
| author | Gao, Jinkai Ma, Shiwang |
| author_facet | Gao, Jinkai Ma, Shiwang |
| contents | In this paper, we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{2^*-1}+\varepsilon u^{q-1},\quad u>0, &{\text{in}~Ω},\\ \quad \ \ u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $Ω\subset \mathbb R^N$ with $N\ge 3$ is a bounded domain, $q\in(2,2^*)$ and $2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. It is well-known (H. Brézis and L. Nirenberg, \newblock {\em Comm. Pure Appl. Math.}, 36(4):437--477, 1983) that the above problem admits a positive least energy solution for all $\varepsilon >0$ and $q>\max\{2,\frac{4}{N-2}\}$. In the present paper, we first analyze the asymptotic behavior of the positive least energy solution as $\varepsilon\to 0$ and establish a sharp asymptotic characterisation of the profile and blow-up rate of the least energy solution. Then, we prove the uniqueness and nondegeneracy of the least energy solution under some mild assumptions on domain $Ω$. The main results in this paper can be viewed as a generalization of the results for $q=2$ previously established in the literature. But the situation is quite different from the case $q=2$, and the blow-up rate not only heavily depends on the space dimension $N$ and the geometry of the domain $Ω$, but also depends on the exponent $q\in(\max\{2,\frac{4}{N-2}\}, 2^*)$ in a non-trivial way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case Gao, Jinkai Ma, Shiwang Analysis of PDEs In this paper, we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{2^*-1}+\varepsilon u^{q-1},\quad u>0, &{\text{in}~Ω},\\ \quad \ \ u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $Ω\subset \mathbb R^N$ with $N\ge 3$ is a bounded domain, $q\in(2,2^*)$ and $2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. It is well-known (H. Brézis and L. Nirenberg, \newblock {\em Comm. Pure Appl. Math.}, 36(4):437--477, 1983) that the above problem admits a positive least energy solution for all $\varepsilon >0$ and $q>\max\{2,\frac{4}{N-2}\}$. In the present paper, we first analyze the asymptotic behavior of the positive least energy solution as $\varepsilon\to 0$ and establish a sharp asymptotic characterisation of the profile and blow-up rate of the least energy solution. Then, we prove the uniqueness and nondegeneracy of the least energy solution under some mild assumptions on domain $Ω$. The main results in this paper can be viewed as a generalization of the results for $q=2$ previously established in the literature. But the situation is quite different from the case $q=2$, and the blow-up rate not only heavily depends on the space dimension $N$ and the geometry of the domain $Ω$, but also depends on the exponent $q\in(\max\{2,\frac{4}{N-2}\}, 2^*)$ in a non-trivial way. |
| title | Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.16505 |