Ground states of the Schrödinger equation coupled with fourth-order gravitation -- Part 1: the case $K_{a, b} \leq 0$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908734588977152 |
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| author | Ramos, Gustavo de Paula |
| author_facet | Ramos, Gustavo de Paula |
| contents | We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: \[ \begin{cases} - Δu + (V - ω) u + (K_{a, b} \ast u^2) u = 0 &\text{in} ~ \mathbb{R}^3; \\ \|u\|_{\mathscr{L}^2}^2 = μ, \end{cases} \] where \[ K_{a, b} (x) := \frac{1}{|x|} \left( \frac{4}{3} e^{- b |x|} - \frac{1}{3} e^{- a |x|} - 1 \right); \] $0 \leq a, b \leq \infty$; $V$ denotes a singular potential that vanishes at infinity and the unknowns are $ω\in \mathbb{R}$, $u \colon \mathbb{R}^3 \to \mathbb{R}$. This problem is obtained by looking for standing waves of the Schrödinger equation coupled with the nonrelativistic gravitational potential prescribed by a family of fourth-order gravity theories. In this paper, (i) we obtain a complete picture of the existence/nonexistence of ground states of the associated autonomous problem for every possible geometry of $K_{a, b}$, (ii) we obtain conditions that ensure the existence of ground states of the nonautonomous problem when $K_{a, b} \leq 0$ and (iii) we prove that as \[ (a, b) \to (A, B) \in \left\{(0, 0), (\infty, \infty), (0, \infty)\right\}, \] ground states of this problem respectively converge to a ground state of (1) the Schrödinger equation, (2) the Choquard equation and (3) a rescaling of the Choquard equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground states of the Schrödinger equation coupled with fourth-order gravitation -- Part 1: the case $K_{a, b} \leq 0$ Ramos, Gustavo de Paula Analysis of PDEs We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: \[ \begin{cases} - Δu + (V - ω) u + (K_{a, b} \ast u^2) u = 0 &\text{in} ~ \mathbb{R}^3; \\ \|u\|_{\mathscr{L}^2}^2 = μ, \end{cases} \] where \[ K_{a, b} (x) := \frac{1}{|x|} \left( \frac{4}{3} e^{- b |x|} - \frac{1}{3} e^{- a |x|} - 1 \right); \] $0 \leq a, b \leq \infty$; $V$ denotes a singular potential that vanishes at infinity and the unknowns are $ω\in \mathbb{R}$, $u \colon \mathbb{R}^3 \to \mathbb{R}$. This problem is obtained by looking for standing waves of the Schrödinger equation coupled with the nonrelativistic gravitational potential prescribed by a family of fourth-order gravity theories. In this paper, (i) we obtain a complete picture of the existence/nonexistence of ground states of the associated autonomous problem for every possible geometry of $K_{a, b}$, (ii) we obtain conditions that ensure the existence of ground states of the nonautonomous problem when $K_{a, b} \leq 0$ and (iii) we prove that as \[ (a, b) \to (A, B) \in \left\{(0, 0), (\infty, \infty), (0, \infty)\right\}, \] ground states of this problem respectively converge to a ground state of (1) the Schrödinger equation, (2) the Choquard equation and (3) a rescaling of the Choquard equation. |
| title | Ground states of the Schrödinger equation coupled with fourth-order gravitation -- Part 1: the case $K_{a, b} \leq 0$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.22619 |