Assume $\mathbb{H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{\mathbb{H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ it is well-known that we have inequality (Rellich inequality) $$ \Lambda_N \int_{\Omega}\frac{u^2}{|x|^4}\mathrm{d}x \leq \int_\Omega |\Delta u|^2 \, \mathrm{d}x $$ where $\Lambda_N=(\frac{N^2(N-4)^2}{16})$ is optimal constant and also it is known that $$\|u\|^2=\int_\Omega \Big(|\Delta u|^2-\Lambda_N \frac{u^2}{|x|^4}\Big) \, \mathrm{d}x $$ Defines an other norm on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.
My question is this that is these two norm equivalent? To see an improved case of this inequality with reminder term see Corollary 1 of Paper.
I know if I could show that $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ is a hilbert space with the new inner product then this two norms must be equivalent due to Open mapping theorem.
