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

TL;DR

This paper investigates the existence, nonexistence, and asymptotic behavior of ground states for a nonlocal Schr"odinger problem coupled with a fourth-order gravitational potential, covering various parameter regimes and limits.

Contribution

It provides a comprehensive analysis of ground states for the Schr"odinger equation with a nonlocal gravitational term, including existence conditions and asymptotic convergence results.

Findings

Complete characterization of ground state existence for all geometries of the kernel

Conditions for ground state existence in the nonautonomous case when kernel is nonpositive

Asymptotic convergence of ground states to Schr"odinger or Choquard solutions as parameters vary

Abstract

We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: \[ \begin{cases} - \Delta u + (V - \omega) u + (K_{a, b} \ast u^2) u = 0 &\text{in} ~ \mathbb{R}^3; \\ \|u\|_{\mathscr{L}^2}^2 = \mu, \end{cases} \] 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 $\omega \in \mathbb{R}$, $u \colon \mathbb{R}^3 \to \mathbb{R}$. This problem is obtained by looking for standing waves of the Schr\"odinger 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 Schr\"odinger equation, (2) the Choquard equation and (3) a rescaling of the Choquard equation.