摘要
讨论了方程-△ pu =-Div(|Du|p -2 Du) =Q(x) |u|p -2 u +ε|u|σ -1u x∈Ωu|Ω=0的极小能量解在ε→ 0时的形态 :当ε→ 0时 ,方程极小能量解uε 在测度意义下满足|Duε|p 弱 Q- N -ppm SNp δx0 ,|uε|p 弱 Q- Npm SNp δx0 ,其中Qm=maxx∈ΩQ(x) =Q(x0 ) ,δx0 为x0 的Dirac函数 ,Ω是有界光滑区域 .
This paper deals with the shape of the least energy solution of quasilinear elliptic equations involving critical exponents -△ pu=- Div (｜Du｜ p-2 Du)=Q(x)｜u｜ p *-2 u+ε｜u｜ σ-1 u x∈Ω, u｜ Ω =0, where Ω is a bounded domain in R N with smooth boundary Ω,0<ε<λ 1 ,and ε→0.Q(x)∈C(Ω),σ∈[1,p ) .The following conclusions are proved:The least energy solution u ε of the equations satisfies (after passing to a subsequence): ｜Du ε｜ p w Q -N-pp m S Np δ x 0 ,as ε→0,in the sense of measure. ｜u ε｜ p * w Q -Np m S Np δ x 0 ,as ε→0,in the sense of measure. Where Q m = max x∈Ω Q(x)=Q(x 0),δ x 0 be the Dirac mass at x 0 , S is the best Sobolev constant.
基金
湖南省自然科学基金资助项目!(99JJY2 0 0 3)
