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

TL;DR

This work characterizes ground states for a normalized nonlocal Schrödinger equation with a convolution kernel K_{a,b} that models a fourth-order gravity modification. It first settles the autonomous problem: ground states exist if and only if b>0 (with negative energy and negative Lagrange multiplier), while no ground states occur when b=0. It then extends to the nonautonomous setting with a potential V, introducing an energy-deficiency condition E_{a,b}^V(\\mu) < E_{a,b}^0(\\mu) that guarantees ground states under mild integrability/vanishing-at-infinity assumptions and yields ω<0. The paper further analyzes the asymptotic behavior of ground states as (a,b) approach key limits, showing convergence to Schrödinger, Choquard, or rescaled Choquard-type ground states (up to translations in V=0 cases). The results provide a unified framework for existence and asymptotics across physically relevant regimes, with stability under perturbations of the potential and clear connections to classical nonlocal equations.

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 \[ K_{a, b} (x) := \frac{1}{|x|} \left( \frac{4}{3} e^{- b |x|} - \frac{1}{3} e^{- a |x|} - 1 \right); \] $0 \leq a, b \leq \infty$; $V$ 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.

Figures (9)

• Figure 1: $0 \leq a \leq 2 b = \infty$
• Figure 2: $0 \leq a < 2 b < \infty$
• Figure 3: $0 < a = 2 b < \infty$
• Figure 4: $0 < 2 b < a < 4 b < \infty$
• Figure 5: $0 < a = 4 b < \infty$

Theorems & Definitions (26)

• Theorem 1.1
• Example
• Proposition 1.2
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3