Schrödinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour
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arXiv
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| Format: | Preprint |
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2025
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| author | Damian, Heydy M. Santos Siciliano, Gaetano |
| author_facet | Damian, Heydy M. Santos Siciliano, Gaetano |
| contents | We consider the following Schrödinger-Bopp-Podolsky system with critical and sublinear terms \begin{equation*} \begin{cases}
- Δu+ u+Q(x)ϕu= \vert u\vert^4 u+ λK(x)\vert u \vert^{p-1}u&\mbox{ in }\ \mathbb{R}^3 \smallskip
- Δϕ+ a^{2}Δ^{2} ϕ= 4πQ(x) u^{2}& \mbox{ in }\ \mathbb{R}^3. \end{cases} \end{equation*} Here $u,ϕ:\mathbb {R}^{3}\rightarrow \mathbb{R}$ are the unknowns, $Q$ and $K$ are given functions satisfying mild assumptions, $a\geq0, λ>0$ are parameters and $p\in (0,1)$.
We first show existence of infinitely many solutions at negative energy level, including the ground state, when the parameter $λ$ is small. Then
we give general results concerning the structure of the set of solutions.
We show also the behaviour of the solutions as the parameters $a,λ$ tend to zero. In particular the ground states solutions tends to a ground state solution of the Schrödinger-Poisson system as $a$ tends to zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19444 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schrödinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour Damian, Heydy M. Santos Siciliano, Gaetano Analysis of PDEs We consider the following Schrödinger-Bopp-Podolsky system with critical and sublinear terms \begin{equation*} \begin{cases} - Δu+ u+Q(x)ϕu= \vert u\vert^4 u+ λK(x)\vert u \vert^{p-1}u&\mbox{ in }\ \mathbb{R}^3 \smallskip - Δϕ+ a^{2}Δ^{2} ϕ= 4πQ(x) u^{2}& \mbox{ in }\ \mathbb{R}^3. \end{cases} \end{equation*} Here $u,ϕ:\mathbb {R}^{3}\rightarrow \mathbb{R}$ are the unknowns, $Q$ and $K$ are given functions satisfying mild assumptions, $a\geq0, λ>0$ are parameters and $p\in (0,1)$. We first show existence of infinitely many solutions at negative energy level, including the ground state, when the parameter $λ$ is small. Then we give general results concerning the structure of the set of solutions. We show also the behaviour of the solutions as the parameters $a,λ$ tend to zero. In particular the ground states solutions tends to a ground state solution of the Schrödinger-Poisson system as $a$ tends to zero. |
| title | Schrödinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.19444 |