MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS KLEIN-GORDON EQUATION WITH SIGN-CHANGING POTENTIAL COUPLED WITH BORN-INFELD THEORY

Lixia Wang; Chunlian Xiong; Dong Zhang

doi:10.11948/20230034

2024 Volume 14 Issue 1

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Lixia Wang, Chunlian Xiong, Dong Zhang. MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS KLEIN-GORDON EQUATION WITH SIGN-CHANGING POTENTIAL COUPLED WITH BORN-INFELD THEORY[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 84-105. doi: 10.11948/20230034

Citation:

Lixia Wang, Chunlian Xiong, Dong Zhang. MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS KLEIN-GORDON EQUATION WITH SIGN-CHANGING POTENTIAL COUPLED WITH BORN-INFELD THEORY[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 84-105. doi: 10.11948/20230034

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS KLEIN-GORDON EQUATION WITH SIGN-CHANGING POTENTIAL COUPLED WITH BORN-INFELD THEORY

1.
School of Sciences, Tianjin Chengjian University, Jinjing Road No. 26, Tianjin, 300384, China
2.
School of Economics and Management, Tianjin Chengjian University, Jinjing Road No. 26, Tianjin, 300384, China

^*This work was done while the first author was visiting the Chern Institute of Mathematics at Nankai University.
Author Bio: Email: wanglixia0311@126.com(L. X. Wang); Email: zhangdongtg@126.com(D. Zhang)
Corresponding author: Email: xiongcltg@126.com(C. L. Xiong)

Abstract

Abstract

In this paper, we study the following nonhomogeneous Klein-Gordon equation with Born-Infeld theory
$ \begin{align*} \begin{cases} - \Delta u +\lambda V(x)u-K(x)(2\omega+\phi)\phi u =f(x,u)+h(x), &x\in \mathbb{R}^3,\\ \Delta \phi +\beta\Delta_4\phi=4\pi K(x)(\omega+\phi)u^2, \quad & x\in \mathbb{R}^3, \\ \end{cases} \end{align*} $
where $ \omega>0 $ is a constant, $ \lambda>0 $ is a parameter and $ \triangle_4\phi=div(|\nabla\phi|^2\nabla\phi) $. Under some suitable assumptions on $ V,K,f $ and $ h $, the existence of multiple solutions is proved by using the Linking theorem and the Ekeland's variational principle in critical point theory. Especially, the potential $ V $ is allowed to be sign-changing.
- Klein-Gordon equation /
- nonhomogeneous /
- Born-Infeld theory /
- Ekeland's variational principle /
- sign-changing potential
MSC: 35B33, 35J65, 35Q55

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References

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