On the existence of ground states to Hartree-type equations in $\mathbb{R}^3$ with a delta potential

TL;DR

This paper investigates the existence of ground states for a Hartree-type equation with a delta potential in three-dimensional space, establishing conditions under which ground states exist or do not exist based on parameters.

Contribution

It provides a rigorous analysis of ground state existence for the Hartree equation with delta potential, identifying parameter regimes with and without ground states using Pohožaev identities.

Findings

No ground states for critical exponent p = (3 + β)/3 with α ≥ 0.

Existence of ground states for supercritical p in a specified range including p=2, β=2.

Characterization of parameter regimes determining ground state existence.

Abstract

Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential formally described by $$ i \partial_t \psi = - \Delta_x \psi + \alpha \delta_0 \psi - (I_\beta \ast |\psi|^p) |\psi|^{p - 2} \psi $$ where $\alpha \in \mathbb{R}$; $0 < \beta < 3$ and we want to solve for $\psi \colon \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C}$. By means of a Poho\v{z}aev identity, we show that if $p = (3 + \beta) / 3$ and $\alpha \geq 0$, then the problem has no ground state at any mass $\mu > 0$. We also prove that if $$ \frac{3 + \beta}{3} < p < \min \left( \frac{5 + \beta}{3}, \frac{5 + 2 \beta}{4} \right), $$ which includes the physically-relevant case $p = \beta = 2$, then the problem admits a ground state at any mass $\mu > 0$.