Existence and Multiplicity of Solutions for a Biharmonic Kirchhoff Equation in $\mathbb{R}^5$

Abstract

We consider the biharmonic equation $∆^2u− (a+b\int_{\mathbb{R}^5} |∇u|^2 dx) ∆u + V (x)u = f(u),$ 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 $p ∈ (2, 10),$ where it was assumed $p ∈ (4, 10)$ in many papers.