Multiple solutions for the fourth-order Kirchhoff type problems in $  \mathbb{R}^N $ involving concave-convex nonlinearities

Zijian Wu; Haibo Chen; Zijian Wu; Haibo Chen

doi:10.3934/era.2022044

Electronic Research Archive

2022, Volume 30, Issue 3: 830-849. doi: 10.3934/era.2022044

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Research article

Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities

Zijian Wu ,
Haibo Chen ^,

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received: 18 August 2021 Revised: 27 November 2021 Accepted: 28 November 2021 Published: 01 March 2022

In this paper, we study the multiplicity of solutions for the following fourth-order Kirchhoff type problem involving concave-convex nonlinearities and indefinite weight function

$ \begin{equation*} \Delta^2u-\left(a+b\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+V(x)u = \lambda f(x)|u|^{q-2}u+|u|^{p-2}u, \end{equation*} $

where $ u\in H^2(\mathbb{R}^N)(4 < N < 8) $, $ \lambda > 0, 1 < q < 2, 4 < p < 2_\ast(2_\ast = 2N/(N-4)) $, $ f(x) $ satisfy suitable conditions, and $ f(x) $ may change sign in $ \mathbb{R}^N $. Using Nehari manifold and fibering maps, the existense of multiple solutions is established. Moreover, the existence of sign-changing solution is obtained for $ f(x)\equiv0 $. Our results generalize some recent results in the literature.
- fourth-order Kirchhoff type problems,
- multiple solutions,
- indefinite weight functions,
- Nehari manifold,
- fibering maps
Citation: Zijian Wu, Haibo Chen. Multiple solutions for the fourth-order Kirchhoff type problems in $ \mathbb{R}^N $ involving concave-convex nonlinearities[J]. Electronic Research Archive, 2022, 30(3): 830-849. doi: 10.3934/era.2022044

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Abstract

In this paper, we study the multiplicity of solutions for the following fourth-order Kirchhoff type problem involving concave-convex nonlinearities and indefinite weight function

$ \begin{equation*} \Delta^2u-\left(a+b\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+V(x)u = \lambda f(x)|u|^{q-2}u+|u|^{p-2}u, \end{equation*} $

where $ u\in H^2(\mathbb{R}^N)(4 < N < 8) $, $ \lambda > 0, 1 < q < 2, 4 < p < 2_\ast(2_\ast = 2N/(N-4)) $, $ f(x) $ satisfy suitable conditions, and $ f(x) $ may change sign in $ \mathbb{R}^N $. Using Nehari manifold and fibering maps, the existense of multiple solutions is established. Moreover, the existence of sign-changing solution is obtained for $ f(x)\equiv0 $. Our results generalize some recent results in the literature.

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