具变量增长椭圆方程弱解的存在性

The existence of weak solutions for elliptic equations with variable growth

在变指数Sobolev空间W1,p(x)框架下研究了具有变量增长条件的椭圆型偏微分方程-div(a(x,u,u))+g(x,u,u)=f的D irichlet问题,这里Caratheodory函数a(x,s,ξ)具有对ξ的单调性,Caratheodory函数g(x,s,ξ)满足g(x,s,ξ)s≥0.利用逼近论证的方法得到了当f∈W-1,p′(x)空间时在自反的W10,p(x)空间中弱解的存在性。

The Dirichlet problem of elliptic partial differential equations ： - div （a（x,u,↓△u））＋g（x,u,↓△u）=f with variable growth is studied in the setting of variable exponent Sobolev space W^1,p（x） , function a（x ,s ,ξ） is monotone with respect to ξ and the Caratheodory function g（x ,s ,ξ）, where the Caratheodory satisfies g（ x ,s ,ξ ） s ≥10. The existence of weak solutions in reflexive W0^1,p（x） space is obtained by approximation method,if f∈W^-1,p′（x）.