Normalized solutions for a class of nonlinear Choquard equations

We prove the existence of a least energy solution to the problem - Delta u - ( I alpha & lowast; F ( u ) ) f ( u ) = lambda u in R N , integral R N u 2 ( x ) d x = a 2 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta u-(I_{\alpha }*F(u))f(u)=\lambda u\ \text { in }\ {\mathbb {R}}<^>{N},\quad \int _{{\mathbb {R}}<^>N}u<^>2(x)dx = a<^>2, \end{aligned}$$\end{document} where N >= 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 1$$\end{document} , alpha is an element of ( 0 , N ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,N)$$\end{document} , F ( s ) : = integral 0 s f ( t ) d t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(s):=\int _{0}<^>{s}f(t)dt$$\end{document} , and I alpha : R N -> R \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\alpha }:{\mathbb {R}}<^>{N}\rightarrow {\mathbb {R}}$$\end{document} is the Riesz potential. If f is odd in u then we prove the existence of infinitely many normalized solutions.