Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass
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| Format: | Preprint |
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2025
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| _version_ | 1866916746158407680 |
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| author | Lv, Zongyan Zeng, Xiaoyu Zhou, Huan-Song |
| author_facet | Lv, Zongyan Zeng, Xiaoyu Zhou, Huan-Song |
| contents | In this paper, we consider the following Brézis-Nirenberg problem with prescribed $ L^2$-norm (mass) constraint:
\begin{equation*}
\begin{cases}
-Δu=|u|^{2^*-2} u +λ_ρu\quad \text { in } Ω,
u>0, \quad u \in H_0^1(Ω), \quad \int_Ω u^2dx=ρ,
\end{cases}
\end{equation*}
where $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $ρ>0$ is a given small constant and $λ_ρ>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $Ω$. Our results extend those in \cite{BHG3,DGY,SZ}, where the authors obtained one or two positive solutions corresponding to the (local) minimizer or mountain pass type critical point for the energy functional of above equation. Furthermore, using blow-up analysis and local Pohozaev identities arguments, we prove that the $k$-spike solutions are locally unique. Compared to the standard Brézis-Nirenberg problem without the mass constraint, an additional difficulty arises in estimating the error caused by the differences in the Euler-Lagrange multipliers corresponding to different solutions. We overcome this difficulty by introducing novel observations and estimates related to the kernel of the linearized operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass Lv, Zongyan Zeng, Xiaoyu Zhou, Huan-Song Analysis of PDEs In this paper, we consider the following Brézis-Nirenberg problem with prescribed $ L^2$-norm (mass) constraint: \begin{equation*} \begin{cases} -Δu=|u|^{2^*-2} u +λ_ρu\quad \text { in } Ω, u>0, \quad u \in H_0^1(Ω), \quad \int_Ω u^2dx=ρ, \end{cases} \end{equation*} where $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $ρ>0$ is a given small constant and $λ_ρ>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $Ω$. Our results extend those in \cite{BHG3,DGY,SZ}, where the authors obtained one or two positive solutions corresponding to the (local) minimizer or mountain pass type critical point for the energy functional of above equation. Furthermore, using blow-up analysis and local Pohozaev identities arguments, we prove that the $k$-spike solutions are locally unique. Compared to the standard Brézis-Nirenberg problem without the mass constraint, an additional difficulty arises in estimating the error caused by the differences in the Euler-Lagrange multipliers corresponding to different solutions. We overcome this difficulty by introducing novel observations and estimates related to the kernel of the linearized operators. |
| title | Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.14168 |