Infinitely many solutions for a class of Kirchhoff-type equations

Abstract In this article, we consider a class of Kirchhoff-type equations: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> <m:mtd columnalign="left"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>.</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \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}&amp; x\in \Omega ,\\ u=0,\hspace{1.0em}&amp; 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>b</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:math> b\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>b</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> b=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).