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

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2020,（3）
• 页码范围: 539-558
• 页数: 20页
• 语种: 英文

摘要

In this paper,we study the existence and multiplicity of solutions with a prescribed L2-norm for aclass of nonlinear fractional Choquard equations in RN:(-Δ)su-λu =(κα*|u|p)|u|p-2u,where N ≥ 3,s ∈ (0,1),a ∈ (0,N),p ∈ (max{1 + α+2s/N,2},N+α/N-2s) and κα(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∫RN(κα*|u|p)|u|p on the constraints S(c) ={u ∈Hs(RN):||u||2L2(RN) =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).