Entire solutions for a class of elliptic equations involving $p$-biharmonic operator and Rellich potentials

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations $${\mathrm{\Delta }}_p^{2}u=\lambda \mathrm{\mid }x{\mathrm{\mid }}^{-2p}\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u\text{in}{\mathbb{R}}^N,$$ where $N>2p$, $p>1$, $q>p$, $\beta = N - \frac{q}{p}(N-2p)$and$\lambda\in\mathbb R$ is smaller than the Rellich constant.