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

• 作 者: Gongbao Li Xiao Luo
• 出 处: Science China Mathematics
• 发表日期: 0年期
• 关键词: Existence and multiplicity of normalized solutions for a class of fractional Cho
• 基金项目: supported by National Natural Science Foundation of China (Grant Nos. 11371159 and 11771166)

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:（-?）^su-λu=（κ_α^*|u|^p）|u|^p-2u,where N≥3, s∈（0, 1）, α∈（0,N）, p∈（max{1+（α+2s）/N,2},（N+α）/（N-2s）） and κ_α（x）=|x|^α-N. To get such solutions,we look for critical points of the energy functional I（u） =1/2∫_RN|（-?）^s/2u|²-1/2p（κ_α^*|u|^p）|u|^p on the constraints S（c）={u∈H^s（R^N）:‖u‖_L²（R^N）² = c},c > 0.For the value p∈（max{1+（α+2s）/N,2},（N+α）/（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）.