Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 57, pp. 1-23.

Title: Localized nodal solutions for semiclassical nonlinear Kirchhoff equations
       
Author: Lixia Wang (Tianjin Chengjian Univ., Tianjin, China)

Abstract:  
In this article, we consider the existence of localized sign-changing solutions
for the  semiclassical Kirchhoff equation
$$
-(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx) \Delta u+V(x)u
=|u|^{p-2}u, \quad x\in \mathbb{R}^3,\; u\in H^1({\mathbb{R}^3})
$$
where $4<p<2^{\ast}=6$, $\varepsilon>0$ is a small parameter,
V(x) is a positive function that has a local minimum point P.
When $\varepsilon\to 0$, by using a minimax characterization of higher
dimensional symmetric linking structure via the symmetric mountain pass theorem,
we obtain an infinite sequence of localized sign-changing solutions
clustered at the point P.


Submitted April 18, 2022. Published August 02, 2022.
Math Subject Classifications: 35J20, 35J60.
Key Words: Kirchhoff equations; nodal solutions; penalization method.