Electron. J. Differential Equations, Vol. 2024 (2024), No. 18, pp. 1-11.

Existence of high energy solutions for superlinear coupled Klein-Gordons and Born-Infeld equations

Lixia Wang, Pingping Zhao, Dong Zhang

Abstract:
In this article, we study the system of Klein-Gordon and Born-Infeld equations $$\displaylines{ -\Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u), \quad x\in R^3,\cr \Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad x\in R^3, }$$ where \(\Delta_4\phi=\hbox{div}(|\nabla\phi|^2\nabla\phi)\), \(\omega\) is a positive constant. Assuming that the primitive of \(f(x,u)\) is of 2-superlinear growth in \(u\) at infinity, we prove the existence of multiple solutions using the fountain theorem. Here the potential \(V\) are allowed to be a sign-changing function.

Submitted October 30, 2023. Published February 16, 2024.
Math Subject Classifications: 35B33, 35J65, 35Q55.
Key Words: Klein-Gordon equation; Born-Infeld theory; superlinear; fountain theorem.
DOI: 10.58997/ejde.2024.18

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Lixia Wang
School of Sciences
Tianjin Chengjian University
Tianjin 300384, China
email: wanglixia0311@126.com
  Pingping Zhao
School of Sciences
Tianjin Chengjian University
Tianjin 300384, China
email: ppzhao_math@126.com
Dong Zhang
School of Sciences
Tianjin Chengjian University
Tianjin 300384, China
email: zhangdongtg@126.com

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