Multiplicity results for a fourth-order elliptic equation of <i>p</i>(<i>x</i>)-kirchhoff type with weights

Abstract This paper deals with a class of fourth-order elliptic equations of p ( x )-Kirchhoff type of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mtable columnalign="left left" rowspacing=".1em" columnspacing="1em"> <m:mtr> <m:mtd> <m:msubsup> <m:mi mathvariant="normal">Δ</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msubsup> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>M</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mrow> <m:munder> <m:mo>∫</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:munder> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi mathvariant="normal">∇</m:mi> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mi mathvariant="normal">d</m:mi> </m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mi>W</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mtd> <m:mtd> <m:mtext> in </m:mtext> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mtd> <m:mtd> <m:mtext> on </m:mtext> <m:mi mathvariant="normal">∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:math> $$\begin{cases}\Delta_{p(x)}^2 u-M\left(\int\limits_{\Omega} \frac{1}{p(x)}|\nabla u|^{p(x)} \mathrm{d} x\right) \Delta_{p(x)} u=\lambda W(x)|u|^{q(x)-2} u &amp; \text { in } \Omega, \\ u=\Delta u=0 &amp; \text { on } \partial \Omega,\end{cases}$$ where Ω ⊂ ℝ N ( N ≥ 2) is a smooth bounded domain with boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo>:</m:mo> <m:mover> <m:mi mathvariant="normal">Ω</m:mi> <m:mo accent="false">‾</m:mo> </m:mover> <m:mo stretchy="false">→</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> $\partial \Omega, p: \overline{\Omega} \rightarrow \mathbb{R}$ is a log-Hölder continuous function, M ( t ) = a + bt κ is a Kirchhoff function with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>κ</m:mi> <m:mo>&gt;