Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth

Abstract In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -biharmonic operator: <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:mi>M</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:mi>Ω</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msubsup> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>μ</m:mi> </m:mrow> </m:msup> <m:mo>⁎</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi mathvariant="normal">in</m:mi> <m:mspace width="0.33em"/> <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:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi mathvariant="normal">on</m:mi> <m:mspace width="0.33em"/> <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}{l}M\left(\mathop{\displaystyle \int }\limits_{\Omega }{| \Delta u| }^{p}{\rm{d}}x\right){\Delta }_{p}^{2}u-{\Delta }_{p}u=\lambda \left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u+{| u| }^{{p}^{* }-2}u+f,\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\\ u=\Delta u=0,\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="n