一类非局部问题解的存在性与多重性

Existence and Multiplicity of Solutions for a Class of Nonlocal Problems

考虑一类非局部问题{-(a-b integral from Ω|▽u|~2dx)Δu=λg(x)x∈Ω u=0 x∈Ω其中a>0,b>0,ΩR^N是有界开集,λ>0且g∈H^(-1)(Ω)\{0},这里H^(-1)(Ω)是Sobolev空间H_0~1(Ω)的对偶空间.应用Ekeland变分原理和山路引理证明了:存在λ_*>0,使得:(ⅰ)当λ∈(0,λ_*)时,该非局部问题至少有3个不同的解;(ⅱ)当λ=λ_*时,该非局部问题至少有2个不同的解;(ⅲ)当λ>λ_*时,该非局部问题至少有1个解.

Consider a class of nonlocal problems {-（a-b integral from Ω｜▽u｜~2 dx）Δu=λg（x） x ∈Ω u=0 x ∈Ω where a0,b0,ΩR^N is a bounded open set,λ0 and g∈H^（-1）（Ω）／{0}.The Ekeland＇s variational principle and the mountain pass lemma are applied to proved that there exists λ_＊ 0 such that （ⅰ） The problem has at least three solutions if λ∈（0,λ_＊）;（ⅱ） The problem has at least two solutions if λ=λ_＊;（ⅲ） The problem has at least one solution if λλ_＊.