Normalized Ground States for the Schrödinger Equation with Hartree Type and Square-Root Nonlinearities

We consider the following Schrödinger equation with combined Hartree type and square-root nonlinearities - ▵ u = λ u + μ I α ∗ | u | p | u | p - 2 u + 1 - 1 1 + u 2 u , in R N , having prescribed mass ∫ R N | u | 2 = c , where N ≥ 2 and c > 0 is a given real number, λ appears as a Lagrange multiplier. Under the assumption of μ > 0 and N + α N < p < N + α + 2 N , we prove the existence of the normalized ground state by combining Concentration-compactness principle and estimate on the square-root nonlinearity. The main results extend and complement the earlier works.