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

Consider the Hartree-type equation in ${\mathbb{R}}^3$ with a delta potential formally described by $$i{\mathrm{\partial }}_t\psi =-{\mathrm{\Delta }}_x\psi +\alpha {\delta }_0\psi -(I_{\beta }\ast \mathrm{\mid }\psi {\mathrm{\mid }}^p)\mathrm{\mid }\psi {\mathrm{\mid }}^{p-2}\psi$$ where $\alpha \in \mathbb{R}$; $0<\beta <3$ and we want to solve for $\psi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\times \mathbb{R}\to \mathbb{C}$. By means of a Pohožaev identity, we show that if $p=(3+\beta )\mathrm{/}3$ and $\alpha \ge 0$, then the problem has no ground state at any mass $\mu >0$. We also prove that if $$\frac{3+\beta }{3}<p<\min (\frac{5+\beta }{3},\frac{5+2\beta }{4}),$$ which includes the physically-relevant case $p=\beta =2$, then the problem admits a ground state at any mass $\mu >0$.