Ground states of the Schrödinger equation coupled with fourth-order gravitation -- Part 1: the case $K_{a,b}\le 0$

Abstract

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 \[ K_{a, b} (x) := \frac{1}{|x|} \left( \frac{4}{3} e^{- b |x|} - \frac{1}{3} e^{- a |x|} - 1 \right); \] $0\le a,b\le \infty$; $V$ denotes a singular potential that vanishes at infinity and the unknowns are $\omega \in \mathbb{R}$, $u:{\mathbb{R}}^3\to \mathbb{R}$. 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 $K_{a,b}\le 0$ 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.