| Abstract: |
| A novel approach is exploited to find solutions $(\lambda,u) \in \mathbb{R} \times H^m(\mathbb{R}^N)$ to
\[\begin{cases}
(-\Delta)^m u + \frac{\mu}{|y|^{2m}} u + \lambda u = g(u)\text{,}\
\int_{\mathbb{R}^N}u^2\,\mathrm{d}x=\rho^2\text{,}
\end{cases}\]
where $\mathbb{R}^N \ni x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}$, $N \ge K \ge 2m$, $1 \le m \in \mathbb{N}$, $\rho>0$, and the growth of $g\colon\mathbb{R}\to\mathbb{R}$ is mass-supercritical and Sobolev-subcritical at infinity and at least mass-critical at the origin.
An important step in this approach is to show that all the solutions to the differential equation above that are radial in $y$ satisfy the \textit{Poho\v{z}aev identity}, which in the presence of a Hardy-type potential is only known in the case $m=1$ and $N=K$. |
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