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

Abstract

In this paper, we study the existence and multiplicity of solutions with a prescribed L²-norm for a class of nonlinear fractional Choquard equations in ℝ_N:

$${( - \Delta )^s}u - \lambda u = ({\kappa _a}*|u{|^{p - {2_u}}})$$

where N ⩾ 3, s ∈ (0, 1), α ∈ (0, N), $p \in (max\{ 1 + \frac{{a + 2s}}{N},2\} \frac{{N + a}}{{N - 2s}})and{\kappa _a}(x) = |x{|^{a - N}}$. To get such solutions, we look for critical points of the energy functional

$$I(u) = \frac{1}{2}{\int_{{R^N}} {|{{( - \Delta )}^{\frac{s}{2}}}u|} ^2} - \frac{1}{{2p}}\int_{{R^N}} {({\kappa _a}*|u{|^p})} |u{|^p}$$

on the constraints

$$S(c)\{ u \in {H^S}({R^N}):||u||_{{L^2}({R^N})}^2 = c\} ,c > 0.$$

For the value $p \in (max\{ 1 + \frac{{a + 2s}}{N},2\} ,\frac{{N + a}}{{N - 2s}})$ 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).