One solution to a fourth-order Leray–Lions problem involving p ( x )-Hardy potential

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">D</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> D⊂ RN ( N ≥ 2) be a smooth bounded domain. Consider a class of fourth-order problems involving a p ( x )-Hardy potential of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mi mathvariant="normal">Δ</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mo>=</m:mo> <m:mi>μ</m:mi> <m:mi>θ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>δ</m:mi> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mtd> <m:mtd columnalign="left"> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:mi mathvariant="script">D</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:mtd> <m:mtd columnalign="left"> <m:mtext> </m:mtext> <m:mtext>on</m:mtext> <m:mspace width="0.3333em"/> <m:mi>∂</m:mi> <m:mi mathvariant="script">D</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $${cases}(a(x,{Δ}u))=μ θ (x){ u{ }ᵖ⁽ˣ⁾⁻²u}{δ {(x)}²ᵖ⁽ˣ⁾}+λ f(x,u), & in D, \\ u={Δ}u=0, & on ∂ D, {cases}$$ here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Δ</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:mrow>