In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation

where \(a, b> 0\), \(\lambda \) is unknown and appears as a Lagrange multiplier, \(K\in {\mathcal {C}}({\mathbb {R}}^3, {\mathbb {R}}^+)\) with \(0<\lim _{|y|\rightarrow \infty }K(y)\le \inf _{{\mathbb {R}}^3} K\), and \(f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) satisfies general \(L^2\)-supercritical or \(L^2\)-subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f. This paper extends to the nonautonomous case previous results on prescribed \(L^2\)-norm solutions of Kirchhoff problems.

在本文中，我们建立了以下 Kirchhoff 型方程的归一化解的存在性

其中\(a, b> 0\) , \(\lambda \)是未知的，并表现为一个拉格朗日乘数，\(K\in {\mathcal {C}}({\mathbb {R}}^3, { \mathbb {R}}^+)\)与\(0<\lim _{|y|\rightarrow \infty }K(y)\le \inf _{{\mathbb {R}}^3} K\ )，和\(f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\)满足一般\(L^2\) -超临界或\(L^ 2\) -亚临界条件。我们引入了一些新的分析技术，以排除由于潜在K的存在和非线性f的同质性的缺乏而导致的最小化序列的消失和二分法情况。. 本文扩展到非自治情况下先前关于规定的基尔霍夫问题的\(L^2\) -范数解的结果。