Sign-changing multi-peak standing waves of the NLSE with a point interaction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912313256181760 |
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| author | Ramos, Gustavo de Paula |
| author_facet | Ramos, Gustavo de Paula |
| contents | Consider the following semilinear problem with a point interaction in $\mathbb{R}^N$: \[- Δ_αu + ωu = u |u|^{p - 2},\] where $N \in \{2, 3\}$; $ω> 0$; $- Δ_α$ denotes the Hamiltonian of point interaction with inverse $s$-wave scattering length $- (4 πα)^{- 1}$ and we want to solve for $u \colon \mathbb{R}^N \to \mathbb{R}$. By means of Lyapunov--Schmidt reduction, we prove that this problem has sign-changing multi-peak solutions when either (1) $N = 2$, $α\in \mathbb{R}$, $p_* < p \leq 3$ and $ω$ is sufficiently large or (2) $N = 3$, $0 < α< \infty$, $p_* < p < 3$ and $ω$ is sufficiently small, where $2.45 < p_* := \frac{9 + \sqrt{113}}{8} < 2.46$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sign-changing multi-peak standing waves of the NLSE with a point interaction Ramos, Gustavo de Paula Analysis of PDEs Consider the following semilinear problem with a point interaction in $\mathbb{R}^N$: \[- Δ_αu + ωu = u |u|^{p - 2},\] where $N \in \{2, 3\}$; $ω> 0$; $- Δ_α$ denotes the Hamiltonian of point interaction with inverse $s$-wave scattering length $- (4 πα)^{- 1}$ and we want to solve for $u \colon \mathbb{R}^N \to \mathbb{R}$. By means of Lyapunov--Schmidt reduction, we prove that this problem has sign-changing multi-peak solutions when either (1) $N = 2$, $α\in \mathbb{R}$, $p_* < p \leq 3$ and $ω$ is sufficiently large or (2) $N = 3$, $0 < α< \infty$, $p_* < p < 3$ and $ω$ is sufficiently small, where $2.45 < p_* := \frac{9 + \sqrt{113}}{8} < 2.46$. |
| title | Sign-changing multi-peak standing waves of the NLSE with a point interaction |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.04960 |