We study the existence and multiplicity of weak solutions for a Kirchhoff–Schrödinger type problem in \(\mathbb R^4\) involving a critical nonlinearity and a suitable small perturbation. When \(N=4\), the Sobolev exponent is \(2^*=4\) and, as a consequence, there is a tie between the growth for the nonlocal term and critical nonlinearity. Such behaviour causes new difficulties to treat our study from an exclusively variational point of view, besides those already known for the local operators. Some tools we used in this paper are the mountain-pass and Ekeland’s Theorems and the Lions’ Concentration Compactness Principle.

中文翻译：

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$$\mathbb R^4$$ R 4 中涉及消失势的非齐次临界基尔霍夫-薛定谔型方程

我们研究了\(\mathbb R^4\)中基尔霍夫-薛定谔型问题的弱解的存在性和多重性，涉及临界非线性和合适的小扰动。当\(N=4\) 时，Sobolev 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。除了本地运营商已经知道的那些行为之外，这种行为会导致新的困难从完全变分的角度来处理我们的研究。我们在本文中使用的一些工具是山口和埃克兰定理以及狮子的浓度紧致原理。