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

Abstract In this paper, we consider a boundary problem for a p -biharmonic equation Δ Δ u p − 2 Δ u = a x u p − 2 u ⁡ ln u + b x u p − 2 u ${\Delta}\left({\left\vert {\Delta}u\right\vert }^{p-2}{\Delta}u\right)=a\left(x\right){\left\vert u\right\vert }^{p-2}u\mathrm{ln}\left\vert u\right\vert +b\left(x\right){\left\vert u\right\vert }^{p-2}u$ with a sign-changing logarithmic nonlinearity a x u p − 2 u ⁡ ln u $a\left(x\right){\left\vert u\right\vert }^{p-2}u\mathrm{ln}\left\vert u\right\vert $ and a polynomial nonlinearity b x u p − 2 u $b\left(x\right){\left\vert u\right\vert }^{p-2}u$ in a bounded domain. We derive a new result regarding the existence of solutions and illustrate how the weight functions a x $a\left(x\right)$ and b x $b\left(x\right)$ can affect the existence and multiplicity of solutions. Furthermore, the sign-changing weight function a x