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 \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^m u+\frac{\mu }{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb {R}^K \times \mathbb {R}^{N-K}, \\ \int _{\mathbb {R}^N} |u|^2 \, dx = \rho > 0, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{(-\mathrm{\Delta })}^{m}u+\frac{\mu }{{|y|}^{2m}}u+\lambda u=g(u),x=(y,z)\in {\mathbb{R}}^K\times {\mathbb{R}}^{N-K},\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=\rho >0,\hfill \end{array}\right)\end{array}$$ in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the \(L^2\) $L^2$ -ball. Moreover, we find also a solution to the related curl–curl problem \(\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times \nabla \times \textbf{U}+\lambda \textbf{U}=f(\textbf{U}), \quad x \in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|\textbf{U}|^2\,dx=\rho ,\\ \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{U}+\lambda \mathbf{U}=f(\mathbf{U}),x\in {\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|\mathbf{U}|}^{2}dx=\rho ,\hfill \end{array}\right)\end{array}$$ which arises from the system of Maxwell equations and is of great importance in nonlinear optics.