In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

中文翻译：

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具有临界指数的基尔霍夫型方程的临界群和多重解

在本文中，通过莫尔斯理论，我们将研究带有附加临界非线性项的基尔霍夫型方程，主要结果是计算临界组，包括零为山口解且非线性在零处共振的情况。作为一种应用，在适当的假设下研究了该方程的非平凡解的多重性，该方程的参数跨越第一个特征值。据我们所知，即使对于具有亚临界非线性的 Kirchhoff 型方程，我们的临界组的估计也是新的。