Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation $$-\mathrm{\Delta }u+u=(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^{\frac{\alpha }{N}+1})\mathrm{\mid }u{\mathrm{\mid }}^{\frac{\alpha }{N}-1}u+f(x,u)\text{ in }{\mathbb{R}}^N$$ where $N\ge 1$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$, the exponent $\frac{\alpha }{N}+1$ is critical with respect to the Hardy–Littlewood–Sobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions.