分数阶p-拉普拉斯问题在有拓扑结构的域上的多解

Multiple Solutions of Fractional p-Laplacian Problems with the Topology of Domain

本文研究了分数阶p-拉普拉斯问题{(-△)_(p)^(s)u=μ|u|^(q-2)u+|u|^(p^(*)_(s)-2_(u)),x∈Ω,u=0,x∈R^(N)/Ω,其中Ω■R^(N)是有连续边界的有界开区域,N>ps,s∈(0,1),(一△)_(p)^(s)是分数阶p-拉普拉斯算子,μ是正的实参数,1<q<∞,q∈[p,p^(*)_(s)),p^(*)_(s)=Np/N-ps是分数阶Sobolev临界指数.本文应用Lusternik-Schnirelmann定理,证明了当q=p,N≥p^(2)s或q∈(p,p^(*)_(s)),N>(p(q+1)s)/(q-p+1)时,分数阶p-拉普拉斯问题在有拓扑结构的有界开区域上至少存在catΩ(Ω)个非平凡解.

In this paper we are concerned with the multiplicity of solutions for the following fractional p-Laplacian problem{(-△)_(p)^(s)u=μ|u|^(q-2)u+|u|^(p^(*)_(s)-2_(u)),x∈Ω,u=0,x∈R^(N)/Ω,whereΩ?R^(N)is an open bounded domain with continuous boundary,N>ps with s∈(0,1),(-Δ)_(p)^(s)is the fractional p-Laplacian operator,μis a positive real parameter,1<q<∞,q∈[p,p^(*)_(s))and p^(*)_(s)=Np/N-ps is the fractional critical Sobolev exponent.Using the LusternikSchnirelmann theory,we show that the problem has at least catΩ(Ω)nontrivial solutions with the topology ofΩprovided q=p and N≥p^(2)s or q∈(p,p^(*)_(s))amd N>(p(q+1)s)/(q-p+1)