On a zero mass Schrödinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908552146190336 |
|---|---|
| author | Pomponio, Alessio Yang, Lianfeng |
| author_facet | Pomponio, Alessio Yang, Lianfeng |
| contents | In this paper, we consider the following zero mass Schrödinger-Bopp-Podolsky system
\[
\begin{cases}
-Δu +q^2ϕu=|u|^{p-2}u,
-Δϕ+a^2Δ^2ϕ=4πu^2,
\end{cases}
\text{ in } \mathbb{R}^3,
\]
where $a>0$ and $q\ne 0$. We complete the study initiated in [2], which relied on a perturbation argument to establish the existence of weak solutions. Here, in contrast, our approach, based on the Mountain Pass Theorem and the splitting lemma, directly yields a ground state solution for $p \in (4,6)$.
Moreover, by deriving a Pohozaev identity, we further obtain some nonexistence results for suitable $p$. Finally, based on the minimax characterization, we also analyse, in the radial case, the asymptotic behaviour of the solutions obtained as $a\to 0$, thereby establishing a link with the zero mass Schrödinger-Poisson system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17479 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a zero mass Schrödinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour Pomponio, Alessio Yang, Lianfeng Analysis of PDEs 35J48, 35J50, 35Q60 In this paper, we consider the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2ϕu=|u|^{p-2}u, -Δϕ+a^2Δ^2ϕ=4πu^2, \end{cases} \text{ in } \mathbb{R}^3, \] where $a>0$ and $q\ne 0$. We complete the study initiated in [2], which relied on a perturbation argument to establish the existence of weak solutions. Here, in contrast, our approach, based on the Mountain Pass Theorem and the splitting lemma, directly yields a ground state solution for $p \in (4,6)$. Moreover, by deriving a Pohozaev identity, we further obtain some nonexistence results for suitable $p$. Finally, based on the minimax characterization, we also analyse, in the radial case, the asymptotic behaviour of the solutions obtained as $a\to 0$, thereby establishing a link with the zero mass Schrödinger-Poisson system. |
| title | On a zero mass Schrödinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour |
| topic | Analysis of PDEs 35J48, 35J50, 35Q60 |
| url | https://arxiv.org/abs/2509.17479 |