Construction of solutions for a critical elliptic system of Hamiltonian type
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912727260200960 |
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| author | Guo, Yuxia Xuanyuan, Congzheng Yuan, Tingfeng |
| author_facet | Guo, Yuxia Xuanyuan, Congzheng Yuan, Tingfeng |
| contents | We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*}
\begin{cases}
-Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline
-Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N,
\end{cases} \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$, $V(|y'|, y'') \not\equiv 0$ is a bounded, nonnegative function on $\mathbb{R}_+ \times \mathbb{R}^{N-2}$ and $p, q > 1$ lie on the critical hyperbola: \[
\frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}. \] By applying the finite-dimensional reduction method and local Pohozaev identities combined with the Green representation formula and technical analysis, we show that, under the assumptions that $N \ge 5$, $(p,q)$ lies in a certain admissible range, and $r^2 V(r, y'')$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of solutions for a critical elliptic system of Hamiltonian type Guo, Yuxia Xuanyuan, Congzheng Yuan, Tingfeng Analysis of PDEs We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*} \begin{cases} -Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$, $V(|y'|, y'') \not\equiv 0$ is a bounded, nonnegative function on $\mathbb{R}_+ \times \mathbb{R}^{N-2}$ and $p, q > 1$ lie on the critical hyperbola: \[ \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}. \] By applying the finite-dimensional reduction method and local Pohozaev identities combined with the Green representation formula and technical analysis, we show that, under the assumptions that $N \ge 5$, $(p,q)$ lies in a certain admissible range, and $r^2 V(r, y'')$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large. |
| title | Construction of solutions for a critical elliptic system of Hamiltonian type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.11251 |