含次临界Sobolev指数半线性合作椭圆方程组非平凡解的存在性

Existence of Nontrivial Solutions of the Semilinear Cooperative Elliptic Systems Involving Subcritical Sobolev Exponents

研究了含次临界Sobolev指数的半线性合作椭圆型方程组,在不同情况下得到了方程组非平凡解的存在性.当在0<λ<λ_1时,定义能量泛函J(u,v)以及Nehari流形N_λ.首先说明泛函J(u,v)有下界,且有一个极小值点,于是在Nehari流形里存在临界点,进而说明方程组在Banach空间E中存在非平凡解.当λ_k<λ<λ_(k+1)时,泛函满足局部环绕定理中的条件,于是得到方程组至少存在一个非平凡解的结论.

One cooperative semilinear elliptic systems involving subcritical Sobolev exponents was investigated. The existence of nontrivial solution to the systems was obtained under different cases. In the case 0〈λ〈λ1, through defining the functional and the corresponding Nehari manifold, we got that the functional J （u,v） was bounded below. Then it proved that the functional had a minimizer, so the functional had a nontrivial critical point in the Nehari manifold and the problem had a nontrivial solution in E. In the case λk〈λ〈λk＋1, the functional satisfied the conditions of local linking theorem. It draws the conclusion that there exists at least one nontrivial solution.