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=λu+μIα∗|u|p|u|p-2u+1-11+u2u,inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\triangle u=\lambda u+\mu \left( I_{\alpha }*|u|^{p}\right) |u|^{p-2}u +\left( 1-\frac{1}{\sqrt{1+u^{2}}}\right) u,\ \ \ \ \ \ \ \textrm{in} \ {\mathbb {R}}^{N}, \end{aligned}$$\end{document}having prescribed mass ∫RN|u|2=c,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{{\mathbb {R}}^{N}}|u|^2 =c,$$\end{document} where N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document} and c>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>0$$\end{document} is a given real number, λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} appears as a Lagrange multiplier. Under the assumption of μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0$$\end{document} and N+αN<p<N+α+2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}$$\end{document}, 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.