Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $  \mathbb{R}^3 $

Kun Cheng; Li Wang; Kun Cheng; Li Wang

doi:10.3934/math.2022922

AIMS Mathematics

2022, Volume 7, Issue 9: 16787-16810. doi: 10.3934/math.2022922

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Research article

Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $

Kun Cheng ^{1
,
,},
Li Wang ²

1.
School of Information Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, China
2.
College of Science, East China Jiaotong University, Nanchang 330013, China

Received: 23 May 2022 Revised: 05 July 2022 Accepted: 07 July 2022 Published: 13 July 2022
MSC : 35J20, 35J65

This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system:

$ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), & x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2, & x \in \mathbb{R}^{3}, \end{array}\right. \end{equation*} $

where $ V, K $ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $ f $, we develop a direct approach to establish the existence of infinitely many nodal solutions $ \{u_k^{b, \lambda}\} $ with a prescribed number of nodes $ k $, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $ \{u_k^{b, \lambda}\} $ is strictly increasing in $ k $, and give a convergence property of $ \{u_k^{b, \lambda}\} $ as $ b\rightarrow 0 $ and $ \lambda \rightarrow 0 $.
- nodal solutions,
- Kirchhoff type problem,
- Schrödinger-Poisson system,
- variational methods,
- disc's theorem
Citation: Kun Cheng, Li Wang. Nodal solutions for the Kirchhoff-Schrödinger-Poisson system in $ \mathbb{R}^3 $[J]. AIMS Mathematics, 2022, 7(9): 16787-16810. doi: 10.3934/math.2022922

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Abstract

This paper is dedicated to studying the following Kirchhoff-Schrödinger-Poisson system:

$ \begin{equation*} \left\{\begin{array}{ll} - \left(a+b \int_{ \mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u+V(|x|) u+\lambda\phi u = K(|x|)f(u), & x \in \mathbb{R}^{3}, \\ -\Delta \phi = u^2, & x \in \mathbb{R}^{3}, \end{array}\right. \end{equation*} $

where $ V, K $ are radial and bounded away from below by positive numbers. Under some weaker assumptions on the nonlinearity $ f $, we develop a direct approach to establish the existence of infinitely many nodal solutions $ \{u_k^{b, \lambda}\} $ with a prescribed number of nodes $ k $, by using the Gersgorin disc's theorem, Miranda theorem and Brouwer degree theory. Moreover, we prove that the energy of $ \{u_k^{b, \lambda}\} $ is strictly increasing in $ k $, and give a convergence property of $ \{u_k^{b, \lambda}\} $ as $ b\rightarrow 0 $ and $ \lambda \rightarrow 0 $.

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