摘要
该文考虑分数阶临界Choquard方程{(−Δ)^(s)u=λ|u|^(q−2)u+(∫_(Ω)|u(y)|^(2∗)μ,s|x−y|^(μ)dy|u|2^(∗)μ,s^(−2)u,u=0,x∈Ω,x∈R^(N)∖Ω(0.1)多解的存在性,其中Ω⊂R^(N)是具有光滑边界的有界开集,N>2s,s∈(0,1),0<μ<N,λ是正实参数,q∈[2,2^(∗)_(s)),2_(s)^(∗)=2N/N−2s是分数阶临界Sobolev指数,2_(μ,s)^(*)=2N-μ/N-2s是Hardy-Littlewood-Sobolev不等式意义下的临界指数.利用Lusternik-Schnirelman定理,证明了当q=2且N≥4N≥4或q∈(2,2_(s)^(∗))且N>2s(q+2)/q时,存在λ^(¯)>0,对λ∈(0,λ^(¯),方程至少有cat_(Ω)(Ω)个非平凡解.
In this paper,we are concerned with the multiplicity of solutions for the following fractional Laplacian problem{(−Δ)^(s)u=λ|u|^(q−2)u+(∫_(Ω)|u(y)|^(2∗)_(μ,s)/|x−y|^(μ)dy)|u|2^(∗)_(μ,s)−2u,x∈Ω,u=0,x∈R^(N)∖ΩwhereΩ⊂R^(N)is an open bounded set with continuous boundary,N>2s with s∈(0,1),λis a real parameter,μ∈(0,N)and q∈[2,2_(s)^(∗)),where 2_(s)^(∗)=2N/N−2s,2_(μ,s)^(∗)=2N−μ/N−2s.Using Lusternik-Schnirelman theory,there existsλ^(¯)>0 such that for anyλ∈(0,λ^(¯)),the problem has at least cat_(Ω)(Ω)nontrivial solutions provided that q=2 andN≥4s or q∈(2,2_(s)^(∗))and N>2s(q+2)/q.
作者
陈琳
刘范琴
Chen Lin;Liu Fanqin(School of Mathematics and Statistics,Jiangxi Normal University,Nanchang 330022)
出处
《数学物理学报（A辑）》
CSCD
北大核心
2022年第6期1682-1704,共23页
Acta Mathematica Scientia
