Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials

Zhi-Guo Wu; Wen Guan; Da-Bin Wang

doi:10.3934/cpaa.2022058

Article Contents

2022, Volume 21, Issue 8: 2495-2528. Doi: 10.3934/cpaa.2022058

This issue Previous Article Next Article Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems

Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

^*Corresponding author
^*Corresponding author

Received: September 30, 2021

Revised: December 31, 2021

Early access: March 2022

Published: August 2022

This work is supported by National Natural Science Foundation of China (No. 11961043 and 11561043)

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Abstract

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem

$ \begin{equation*} -(\varepsilon^{2}a+ \varepsilon b\int _{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v+V(x)v = P(x)f(v)\; \; {\rm{in}}\; \mathbb{R}^{3}, \end{equation*} $

where $ \varepsilon $ is a small positive parameter, $ a, b>0 $ and $ V, P\in C^{1}(\mathbb{R}^{3}, \mathbb{R}) $. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential $ P $ is a positive constant, our result generalizes the result obtained in [5] to Kirchhoff problem.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 47J30, 35J20.

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