Morse 临界群在椭圆方程组中的应用

应用Morse 临界群讨论了如下的变分型的非线性椭圆方程组的非平凡解的存在性:(P) -Δu=λ(m11(x)u+m12(x)ν)+n1(x)|u|q-2u+Fu(x,u,ν) x∈Ω-Δν=λ(m21(x)u+m22(x)ν)+n2(x)|ν|q-2ν+Fv(x,u,ν) x∈Ωu|Ω=ν|Ω=0这儿,q∈(1,2), ni(x)可允许变号,这使得本文的结果是新的.

On the morse critical groups of nonlinear elliptic system

Using the morse eritical groups, the nonlinear elliptic system is discussed.(P) -Δu=λ(m11(x)u+m12(x)ν)+n1(x)|u|q-2u+Fu(x,u,ν) in Ω-Δν=λ(m21(x)u+m22(x)ν)+n2(x)|ν|q-2ν+Fv(x,u,ν) in Ωwhere q∈(1,2), the sign of ni(x) is indefinite, which is new for the results.By comparing the morse critical groups of critical points, one obtains the nontrivial solution of (P).