The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.

中文翻译：

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关于高维Kirchhoff型方程的Brezis-Nirenberg问题

本文涉及参数化的基尔霍夫型问题，涉及高维的临界非线性。通过将变分性质与对与该问题相关的能量函数的纤维图的仔细分析结合起来，可以在亚临界扰动的影响下获得解的存在，不存在和多重性。还讨论了纯功率扰动的特殊情况。通过研究Nehari流形，我们将一般情况扩展到更广泛的参数范围。