Infinitely many solutions for a class of Kirchhoff-type equations

Abstract

In this article, we consider a class of Kirchhoff-type equations: −a+b∫Ω∣∇u∣2dxΔu=f(x,u),x∈Ω,u=0,x∈∂Ω.\left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=f\left(x,u),\hspace{1.0em}& x\in \Omega ,\\ u=0,\hspace{1.0em}& x\in \partial \Omega .\end{array}\right. It is a generalization of the classical wave equation and is often used to model wave propagation in various physical media. The nonlocal term b(∫Ω∣∇u∣2dx)Δub\left({\int }_{\Omega }{| \nabla u| }^{2}{\rm{d}}x)\Delta u in the equation causes the variational functional of the equation to have fundamentally different properties from the case of b=0b=0. As far as we know, there are relatively few conclusions regarding infinitely many solutions of this equation. Under weaker assumptions, we obtain that the equation has infinitely many high-energy solutions. And our results extend the conclusions of Mao-Zhang (2009) and Zhang-Perera (2006).

Keywords

• kirchhoff-type equation
• multiple solutions
• high energy solution
• 35j20