Multiple solutions for the p -biharmonic equation with a polynomial nonlinearity and a sign-changing logarithmic nonlinearity

In this paper, we consider a boundary problem for a p -biharmonic equation <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:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> </m:mrow> </m:mfenced> <m:mo>=</m:mo> <m:mi>a</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</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:mi>ln</m:mi> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:math> ${Δ}({ {Δ}u }ᵖ⁻²{Δ}u)=a(x){ u }ᵖ⁻²uln u +b(x){ u }ᵖ⁻²u$ with a sign-changing logarithmic nonlinearity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>a</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</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:mi>ln</m:mi> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:math> a(x) u ᵖ⁻²uln u and a polynomial nonlinearity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>b</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:math> b(x) u ᵖ⁻²u in a bounded domain. We derive a new result regarding the existence of solutions and illustrate how the weight functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>a</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:math> a(x) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>b</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:math> b(x) can affect the existence and multiplicity of solutions. Furthermore, the sign-changing weight function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>a</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:math> <jats:tex-mat