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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.14549 |
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| _version_ | 1866909503002247168 |
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| author | Zhang, Jia Zhang, Weimin |
| author_facet | Zhang, Jia Zhang, Weimin |
| contents | In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: \[ \begin{cases} \begin{aligned} -Δu&=H_v(u, v) \,\quad&&\text{in}~Ω,\\ -Δv&=H_u(u, v) \,\quad&&\text{in}~Ω,\\ u,\,v&=0~~&&\text{on} ~ \partialΩ,\\ \end{aligned} \end{cases} \] where $N\ge 1$, $Ω\subset \mathbb{R}^N$ is a bounded domain and $H\in C^1( \mathbb{R}^2)$ is strictly convex, even and subcritical. We mainly present two results: (i) When $H$ is superlinear, the system has infinitely many solutions, whose energies tend to infinity. (ii) When $H$ is sublinear, the system has infinitely many solutions, whose energies are negative and tend to 0. As a byproduct, the Lane-Emden system under subcritical growth has infinitely many solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14549 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many solutions for elliptic system with Hamiltonian type Zhang, Jia Zhang, Weimin Analysis of PDEs In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: \[ \begin{cases} \begin{aligned} -Δu&=H_v(u, v) \,\quad&&\text{in}~Ω,\\ -Δv&=H_u(u, v) \,\quad&&\text{in}~Ω,\\ u,\,v&=0~~&&\text{on} ~ \partialΩ,\\ \end{aligned} \end{cases} \] where $N\ge 1$, $Ω\subset \mathbb{R}^N$ is a bounded domain and $H\in C^1( \mathbb{R}^2)$ is strictly convex, even and subcritical. We mainly present two results: (i) When $H$ is superlinear, the system has infinitely many solutions, whose energies tend to infinity. (ii) When $H$ is sublinear, the system has infinitely many solutions, whose energies are negative and tend to 0. As a byproduct, the Lane-Emden system under subcritical growth has infinitely many solutions. |
| title | Infinitely many solutions for elliptic system with Hamiltonian type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.14549 |