临界拟线性椭圆方程极小能量解的形态

讨论了方程-?pu= - Div( | Du|p- 2Du)= Q( x ) | u|^p*- 2u+ε | u|^{σ- 1}u x∈Ω u|∂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 u_ε在测度意义下满足| Du_ε|p 弱Q-N- ppm SNp

x0, | uε |p*
弱
Q-Npm
SNp

x0,其中 Qm= maxx

Q( x ) = Q( x 0) ,
x₀为 x₀ 的 Dirac函数, Ω是有界光滑区域.

The Shape of the Least Energy Solution of Quasilinear Elliptic Equations

This paper deals wi th the shape of the least energy solut ion of quasilinear el liptic equations involving criticale xponents- v pu = - Div( | Du|p- 2Du) = Q( x ) | u|p*- 2u+ E | u|R - 1u   x I 8 ,u| 98= 0,where 8 is a bounded domain in RNwith smooth boundary 98, 0< E< K 1, and Ey 0. Q( x ) I C( 8) , RI [ 1, p*) .The fol lowing conclusions are proved:The least energy solution uE of the equat ions satisfies ( after passing to a subse -quence) :| DuE |p   w  Q-N- ppm   SNpDx 0, as Ey 0, in the sense of measure.| uE |p*   w  Q-Npm   SNpD x0, as Ey 0, in the sense of measure.Where Qm= maxx I 8Q( x ) = Q( x 0) , D x 0 be the Dirac mass at x 0, S is the bes t Sobolev cons tant.