Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916803937042432 |
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| author | Ramos, Gustavo de Paula |
| author_facet | Ramos, Gustavo de Paula |
| contents | Consider the following nonlinear Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 Δu + (V + ϕ) u = u |u|^{p-1}; \\ a^2 Δ^2 ϕ- Δϕ= 4 πu^2, \end{cases} $$ where $a, \varepsilon > 0$; $1 < p < 5$; $V \colon \mathbb{R}^3 \to ]0, \infty[$ and we want to solve for $u, ϕ\colon \mathbb{R}^3 \to \mathbb{R}$. By means of Lyapunov-Schmidt reduction, we show that if $K \geq 2$, $z_0$ is a strict local minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and $\varepsilon$ is sufficiently small, then the system has a multipeak cluster solution with $K$ peaks placed at the vertices of a regular convex $K$-gon centered at $z_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09949 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system Ramos, Gustavo de Paula Analysis of PDEs 35J48, 35B40, 35Q40 Consider the following nonlinear Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 Δu + (V + ϕ) u = u |u|^{p-1}; \\ a^2 Δ^2 ϕ- Δϕ= 4 πu^2, \end{cases} $$ where $a, \varepsilon > 0$; $1 < p < 5$; $V \colon \mathbb{R}^3 \to ]0, \infty[$ and we want to solve for $u, ϕ\colon \mathbb{R}^3 \to \mathbb{R}$. By means of Lyapunov-Schmidt reduction, we show that if $K \geq 2$, $z_0$ is a strict local minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and $\varepsilon$ is sufficiently small, then the system has a multipeak cluster solution with $K$ peaks placed at the vertices of a regular convex $K$-gon centered at $z_0$. |
| title | Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system |
| topic | Analysis of PDEs 35J48, 35B40, 35Q40 |
| url | https://arxiv.org/abs/2311.09949 |