On planar Schrodinger-Poisson systems with repulsive interactions in the mass supercritical regime
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911306369466368 |
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| author | Sun, Juntao Yao, Shuai Zhang, He |
| author_facet | Sun, Juntao Yao, Shuai Zhang, He |
| contents | In this paper, we investigate solutions with prescribed $L^{2}$-norm (i.e., prescribed mass) for the planar Schrödinger-Poisson (SP) equation% \begin{equation*} -Δu+λu+α\left( \log |\cdot |\ast |u|^{2}\right) u=|u|^{p-2}u,\ \text{in}\ Ω_{R} , \end{equation*}% where $λ\in \mathbb{R}$ is unknown, $α<0,p>4$ and $Ω_{R} \subseteq \mathbb{R}^{2}$ is a domain. First, we prove that the energy functional $J$ corresponding to the SP equation in $\mathbb{R}^{2}$ is unbounded both above and below on the Pohozaev manifold $\mathcal{P}$; this explains the reason why the minimax level of $J$ is difficult to determine, as referenced in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Second, we establish the existence of a ground state and a high-energy solution, both with positive energy in a large bounded domain $Ω_{R} $, which is a substantial advancement in addressing an open problem proposed in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Finally, we analyze the asymptotic behavior of solutions as the domain $Ω_{R} $ is extended to the entire space $\mathbb{R}^{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On planar Schrodinger-Poisson systems with repulsive interactions in the mass supercritical regime Sun, Juntao Yao, Shuai Zhang, He Analysis of PDEs In this paper, we investigate solutions with prescribed $L^{2}$-norm (i.e., prescribed mass) for the planar Schrödinger-Poisson (SP) equation% \begin{equation*} -Δu+λu+α\left( \log |\cdot |\ast |u|^{2}\right) u=|u|^{p-2}u,\ \text{in}\ Ω_{R} , \end{equation*}% where $λ\in \mathbb{R}$ is unknown, $α<0,p>4$ and $Ω_{R} \subseteq \mathbb{R}^{2}$ is a domain. First, we prove that the energy functional $J$ corresponding to the SP equation in $\mathbb{R}^{2}$ is unbounded both above and below on the Pohozaev manifold $\mathcal{P}$; this explains the reason why the minimax level of $J$ is difficult to determine, as referenced in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Second, we establish the existence of a ground state and a high-energy solution, both with positive energy in a large bounded domain $Ω_{R} $, which is a substantial advancement in addressing an open problem proposed in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Finally, we analyze the asymptotic behavior of solutions as the domain $Ω_{R} $ is extended to the entire space $\mathbb{R}^{2}$. |
| title | On planar Schrodinger-Poisson systems with repulsive interactions in the mass supercritical regime |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.06863 |