Unilateral global interval bifurcation and one-sign solutions for Kirchhoff type problems

Wenguo Shen; Wenguo Shen

doi:10.3934/math.2024953

AIMS Mathematics

2024, Volume 9, Issue 7: 19546-19556. doi: 10.3934/math.2024953

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Unilateral global interval bifurcation and one-sign solutions for Kirchhoff type problems

Wenguo Shen ^,

College of General Education, Guangdong University of Science and Technology, Dongguan 523083, Guangdong, China

Received: 09 June 2023 Revised: 11 September 2023 Accepted: 08 October 2023 Published: 13 June 2024
MSC : 35B32, 35P05

In this paper, we study the following Kirchhoff type problems:

$ \left\{ \begin{array}{l} -(\int_{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u^{3}+g(u, \lambda), \, \, \, \, \, \, \, \, \mathrm{in}\, \, \Omega,\\ u = 0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \mathrm{on}\, \, \partial\Omega, \end{array} \right. $

where $ \lambda $ is a parameter. Under some natural hypotheses on $ g $ and $ \Omega $, we establish a unilateral global bifurcation result from interval for the above problem. By applying the above result, under some suitable assumptions on nonlinearity, we shall investigate the existence of one-sign solutions for a class of Kirchhoff type problems.
- unilateral global interval bifurcation,
- one-sign solutions,
- Kirchhoff type problems
Citation: Wenguo Shen. Unilateral global interval bifurcation and one-sign solutions for Kirchhoff type problems[J]. AIMS Mathematics, 2024, 9(7): 19546-19556. doi: 10.3934/math.2024953

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Abstract

In this paper, we study the following Kirchhoff type problems:

$ \left\{ \begin{array}{l} -(\int_{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u^{3}+g(u, \lambda), \, \, \, \, \, \, \, \, \mathrm{in}\, \, \Omega,\\ u = 0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \mathrm{on}\, \, \partial\Omega, \end{array} \right. $

where $ \lambda $ is a parameter. Under some natural hypotheses on $ g $ and $ \Omega $, we establish a unilateral global bifurcation result from interval for the above problem. By applying the above result, under some suitable assumptions on nonlinearity, we shall investigate the existence of one-sign solutions for a class of Kirchhoff type problems.

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