AI 中文总结
本文研究推广Brezis-Nirenberg问题的半线性椭圆方程组，确定了其非平凡解的存在条件与仅存在平凡解的条件，丰富了非局部临界非线性椭圆问题的理论研究。
AI 中文摘要
本文研究一类非线性椭圆方程的变分系统，该系统推广了经典的Brezis-Nirenberg问题。除了用向量值未知函数替代标量值未知函数外，所考虑的问题在两个方向上推广了Brezis-Nirenberg问题：其一，非线性项是两个具有适当齐次度的齐次函数之和；其二，非线性项中的Sobolev临界项是非局部的。我们确定了所考虑方程组存在非平凡解的条件，以及该方程组仅存在平凡解的条件。
英文摘要
This paper concerns a variational system of nonlinear elliptic equations that generalizes the classical Brezis-Nirenberg problem. In addition to considering vector-valued unknown functions in place of scalar-valued unknown functions, the problem under consideration generalizes the Brezis-Nirenberg problem in two directions. First, the nonlinearity is the sum of two homogeneous functions of suitable homogeneity degrees. Second, the Sobolev-critical term in the nonlinearity is nonlocal. We establish conditions under which a nontrivial solution to the system under consideration is guaranteed to exist and conditions under which the system under consideration admits only the trivial solution.
