This paper is devoted to clarifying the relationship between the classical Morse theory andthe Mountain Pass Lemma via the local linking concept.It is shown that for a C^1-function f with alocal linking,the m-thcritic...This paper is devoted to clarifying the relationship between the classical Morse theory andthe Mountain Pass Lemma via the local linking concept.It is shown that for a C^1-function f with alocal linking,the m-thcritical group is nontrivial,where m is the Morse index.Combined with thebehavior of f at infinity,this result can be used to offer the existence of nontrivial critical points.展开更多
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,ΩR^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...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,ΩR^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.展开更多
文摘This paper is devoted to clarifying the relationship between the classical Morse theory andthe Mountain Pass Lemma via the local linking concept.It is shown that for a C^1-function f with alocal linking,the m-thcritical group is nontrivial,where m is the Morse index.Combined with thebehavior of f at infinity,this result can be used to offer the existence of nontrivial critical points.
文摘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,ΩR^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.
