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

We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: $$\{\begin{array}{ll}-\mathrm{\Delta }u+(V-\omega )u+(K_{a,b}\ast u^2)u=0& \text{in }{\mathbb{R}}^3;\\ \mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{L}}^2}^2=\mu ,\end{array}$$ where $$K_{a,b}(x):=\frac{1}{\mathrm{\mid }x\mathrm{\mid }}(\frac{4}{3}{e}^{-b\mathrm{\mid }x\mathrm{\mid }}-\frac{1}{3}{e}^{-a\mathrm{\mid }x\mathrm{\mid }}-1);$$ $0\le a,b\le \mathrm{\infty }$; $V$ denotes a singular potential that vanishes at infinity and the unknowns are $\omega \in \mathbb{R}$, $u\text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to \mathbb{R}$. 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 $$(a,b)\to (A,B)\in \{(0,0),(\mathrm{\infty },\mathrm{\infty }),(0,\mathrm{\infty })\},$$ 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.