We establish the existence and multiplicity of solutions for a Kirchhoff-type problem in \(\mathbb R^4\) involving a critical and concave–convex nonlinearity. Since in dimension four, the Sobolev critical exponent is \(2^*=4\), there is a tie between the growth of the nonlocal term and the critical nonlinearity. This turns out to be a challenge to study our problem from the variational point of view. Some of the main tools used in this paper are the mountain-pass and Ekeland’s theorems, Lions’ Concentration Compactness Principle and an extension to \(\mathbb R^N\) of the Struwe’s global compactness theorem.

中文翻译：

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$$\mathbb R^4$$ R 4 中的临界凸凹基尔霍夫型方程，涉及可能在无穷大处消失的势

我们建立了\(\mathbb R^4\) 中涉及临界和凹凸非线性的基尔霍夫型问题的解的存在性和多重性。由于在第四维中，Sobolev 临界指数为\(2^*=4\)，因此非局部项的增长与临界非线性之间存在联系。事实证明，从变分的角度研究我们的问题是一个挑战。本文中使用的一些主要工具是山口和 Ekeland 定理、Lions 浓度紧致原理以及Struwe 全局紧致定理\(\mathbb R^N\)的扩展。