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 $$ \Delta_{p}^2 u = \lambda|x|^{-2p}|u|^{p-2}u + |x|^{-\beta}|u|^{q-2} u\quad\text{in} \quad \mathbb R^N, $$ 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.