Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915157909700608 |
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| author | Murcia, Edwin G. Siciliano, Gaetano |
| author_facet | Murcia, Edwin G. Siciliano, Gaetano |
| contents | Given a smooth bounded domain $Ω\subset \mathbb R^3$, we consider the following nonlinear Schrödinger-Poisson type system \begin{equation*}
\left\{
\begin{array}{ll}
-Δu+ ϕu -\abs{u}^{p-2}u = ωu & \quad \text{in } λΩ,
-Δϕ=u^{2}& \quad \text{in }λΩ,
u>0 &\quad \text{in }λΩ,
u =ϕ=0 &\quad \text{on }\partial (λΩ),
\int_{λΩ}u^{2} \,\text{d} x=ρ^2
\end{array}
\right. \end{equation*} in the expanding domain $λΩ\subset \mathbb R^{3}, λ>1$ and $p\in (2,3)$, in the unknowns $(u,ϕ,ω)$.
We show that, for arbitrary large values of the expanding parameter $λ$ and arbitrary small values of the mass $ρ>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $λΩ$. Moreover we show that as $λ\to+\infty$ the solutions found converge to a ground state of the problem in the whole space $\mathbb R^{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour Murcia, Edwin G. Siciliano, Gaetano Analysis of PDEs Given a smooth bounded domain $Ω\subset \mathbb R^3$, we consider the following nonlinear Schrödinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -Δu+ ϕu -\abs{u}^{p-2}u = ωu & \quad \text{in } λΩ, -Δϕ=u^{2}& \quad \text{in }λΩ, u>0 &\quad \text{in }λΩ, u =ϕ=0 &\quad \text{on }\partial (λΩ), \int_{λΩ}u^{2} \,\text{d} x=ρ^2 \end{array} \right. \end{equation*} in the expanding domain $λΩ\subset \mathbb R^{3}, λ>1$ and $p\in (2,3)$, in the unknowns $(u,ϕ,ω)$. We show that, for arbitrary large values of the expanding parameter $λ$ and arbitrary small values of the mass $ρ>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $λΩ$. Moreover we show that as $λ\to+\infty$ the solutions found converge to a ground state of the problem in the whole space $\mathbb R^{3}$. |
| title | Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.12626 |