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

Abstract

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.