全空间中一类椭圆方程组解的存在性

Existence of solutions for elliptic system in the whole space

讨论了下面问题:{-Δu+qu=2αα+β|u|α-2u|v|β,x∈RN,-Δv+qv=2βα+β|u|α|v|β-2v,x∈RN,u,v∈H1(RN)。应用变分法证明了以上椭圆方程组至少存在一个非平凡的非负解。

By using the calculus of variations, we can prove that at least one nontrivia and nonnegative solution exists in the following elliptic system.-Δu＋qu=2αα＋β｜u｜α-2u｜v｜β,x∈RN,-Δv＋qv=2βα＋β｜u｜α｜v｜β-2v,x∈RN,u,v∈H1（RN）.