关于一类合作椭圆系统的正解(英文)

On a Positive Solution for a Class Cooperative Elliptic Systems

通过上下解方法和极大值原理,证明了当ε很小时,椭圆系统-Δu=F/u(x,u,v)+εg(x)x∈Ω-Δv=F/v(x,u,v)+εh(x)x∈Ωu>0,v>0x∈Ωu=v=0x∈Ω的极小正解的存在性,其中Ω是RN上的有界光滑区域;F∈C1(Ω×(R+)2,R+);g,h∈C1(Ω);ε是正参数.此外,也证明了当ε很大时该系统无解.

With the sub-supersolution method and the maximum principle,.the existence of a minimal positive solution is proved for the following system -△u=δF/δu（x,u,v）＋εg（x）,-△v=δF/δv（x,u,v）＋εh（x）inΩ;u,v〉0 in Ω;and u=v=0 on δΩ,with ε sufficiently small, where Ω is a bounded smooth domain in R^N ;F∈C^1（-↑Ω×（R^＋）^2,R^＋）;g,h∈C^1（-↑Ω）;and e is a positive parameter. In addition, it is proved that the problem has no solution for ε large enough.