Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems

We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-Δ)^m u+\fracμ{|y|^{2m}}u + λu = g(u), \quad x = (y,z) \in $\mathbb{R}$^K \times $\mathbb{R}$^{N-K}, \ \int_{$\mathbb{R}$^N} |u|^2 , dx = ρ> 0, \end{cases} \end{equation*} in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the $L^2$-ball. Moreover, we find also a solution to the related curl-curl problem \begin{equation*} \begin{cases} \nabla\times\nabla\times$\mathbf{U}$+λ$\mathbf{U}$=f($\mathbf{U}$), \quad x \in $\mathbb{R}$^N, \ \int_{$\mathbb{R}$^N}|$\mathbf{U}$|^2,dx=ρ, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics.