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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.13806 |
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| _version_ | 1866916217127698432 |
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| author | Ramos, Gustavo de Paula |
| author_facet | Ramos, Gustavo de Paula |
| contents | Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential formally described by $$ i \partial_t ψ= - Δ_x ψ+ αδ_0 ψ- (I_β\ast |ψ|^p) |ψ|^{p - 2} ψ$$ where $α\in \mathbb{R}$; $0 < β< 3$ and we want to solve for $ψ\colon \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C}$. By means of a Pohožaev identity, we show that if $p = (3 + β) / 3$ and $α\geq 0$, then the problem has no ground state at any mass $μ> 0$. We also prove that if $$ \frac{3 + β}{3} < p < \min \left(
\frac{5 + β}{3}, \frac{5 + 2 β}{4} \right), $$ which includes the physically-relevant case $p = β= 2$, then the problem admits a ground state at any mass $μ> 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13806 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of ground states to Hartree-type equations in $\mathbb{R}^3$ with a delta potential Ramos, Gustavo de Paula Analysis of PDEs 35Q55 Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential formally described by $$ i \partial_t ψ= - Δ_x ψ+ αδ_0 ψ- (I_β\ast |ψ|^p) |ψ|^{p - 2} ψ$$ where $α\in \mathbb{R}$; $0 < β< 3$ and we want to solve for $ψ\colon \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C}$. By means of a Pohožaev identity, we show that if $p = (3 + β) / 3$ and $α\geq 0$, then the problem has no ground state at any mass $μ> 0$. We also prove that if $$ \frac{3 + β}{3} < p < \min \left( \frac{5 + β}{3}, \frac{5 + 2 β}{4} \right), $$ which includes the physically-relevant case $p = β= 2$, then the problem admits a ground state at any mass $μ> 0$. |
| title | On the existence of ground states to Hartree-type equations in $\mathbb{R}^3$ with a delta potential |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2404.13806 |