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Existence and Multiplicity of Solutions for a Biharmonic Kirchhoff Equation in $\mathbb{R}^5$
  
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KeyWord:Biharmonic equation, multiplicity of solutions, variational method
Author NameAffiliation
Ziqing Yuan Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, China 
Sheng Liu Big Data College, Tongren University, Tongren, Guizhou 554300, China 
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Abstract:
      We consider the biharmonic equation $\Delta^2u-\left(a+b\int_{\mathbb{R}^5}|\nabla u|^2dx\right)\Delta u\\+V(x)u=f(u)$, where $V(x)$ and $f(u)$ are continuous functions. By using a perturbation approach and the symmetric mountain pass theorem, the existence and multiplicity of solutions for this equation are obtained, and the power-type case $f(u)=|u|^{p-2}u$ is extended to $p\in(2,10)$, where it was assumed $p\in(4,10)$ in many papers.