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−Δu+u=(Iα⁎|u|αN+1)|u|αN−1u+f(x,u) in RN where N≥1, Iα is the Riesz potential of order α∈(0,N), the exponent αN+1 is critical with respect to the Hardy-Littlewood-Sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions.