Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed L-2-norm for a class of nonlinear fractional Choquard equations in Double-struck capital R-N: (-Delta)su-lambda u=(kappa a*|u|p-2u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${( - \Delta )<<^>>s}u - \lambda u = ({\kappa _a}*|u{|<<^>>{p - {2_u}}})$$\end{document} where N > 3, s is an element of (0, 1), alpha is an element of (0, N), p is an element of(max{1+a+2sN,2}N+aN-2s)and kappa a(x)=|x|a-N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (max\{ 1 + \frac{{a + 2s}}{N},2\} \frac{{N + a}}{{N - 2s}})and{\kappa _a}(x) = |x{|<<^>>{a - N}}$$\end{document} considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c > 0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that, we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover, by using a minimax procedure, we prove that for any c > 0, there are infinitely many radial critical points of I restricted on S(c).