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

Abstract

Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential formally described by $$ i \partial_t ψ= - Δ_x ψ+ αδ_0 ψ- (I_β\ast |ψ|^p) |ψ|^{p - 2} ψ$$ where $α\in \mathbb{R}$; $0 < β< 3$ and we want to solve for $ψ\colon \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C}$. By means of a Pohožaev identity, we show that if $p = (3 + β) / 3$ and $α\geq 0$, then the problem has no ground state at any mass $μ> 0$. We also prove that if $$ \frac{3 + β}{3} < p < \min \left( \frac{5 + β}{3}, \frac{5 + 2 β}{4} \right), $$ which includes the physically-relevant case $p = β= 2$, then the problem admits a ground state at any mass $μ> 0$.