The Journal of Geometric Analysis ( IF 1.2 ) Pub Date : 2021-06-19 , DOI: 10.1007/s12220-021-00722-0 Quanqing Li , Jianjun Nie , Wenbo Wang , Jian Zhang
In this paper, we study the existence and asymptotic behavior of localized nodal solutions for the following Kirchhoff-type equation
$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$where \(a>0\), \(b>0\) and \(4<p<6\). Under only a local condition that V has a local trapping potential well, when \(\varepsilon >0\) is sufficiently small, we construct the existence of a sequence of localized nodal solutions concentrating around the local minimum points of the potential function V by using variational method and penalization approach. Moreover, we regard b as a parameter and study the asymptotic behavior of the nodal solutions as \(b\searrow 0\), which reflects some relationship between \(b>0\) and \(b=0\).
中文翻译：
一类基尔霍夫方程局部节点解的存在性和渐近行为
在本文中，我们研究了以下 Kirchhoff 型方程的局部节点解的存在性和渐近行为
$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u= |u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$其中\(a>0\)、\(b>0\)和\(4<p<6\)。下只有一个本地条件是V具有局部捕集势阱中，当\（\ varepsilon> 0 \）足够小，我们构建局部淋巴结溶液浓缩周围的电位函数的局部最小值点的序列的存在V由使用变分法和惩罚法。此外，我们将b作为参数并将节点解的渐近行为研究为\(b\searrow 0\)，这反映了\(b>0\)和\(b=0\)之间的某种关系。
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