Normalized Solutions to at Least Mass Critical Problems: Singular Polyharmonic Equations and Related Curl–Curl Problems

Abstract

AbstractWe are interested in the existence of normalized solutions to the problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^m u+\frac{\mu }{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb {R}^K \times \mathbb {R}^{N-K}, \\ \int _{\mathbb {R}^N} |u|^2 \, dx = \rho > 0, \end{array}\right. } \end{aligned}$$ ( - Δ ) m u + μ | y | 2 m u + λ u = g ( u ) , x = ( y , z ) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , 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 $$\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times \nabla \times \textbf{U}+\lambda \textbf{U}=f(\textbf{U}), \quad x \in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|\textbf{U}|^2\,dx=\rho ,\\ \end{array}\right. } \end{aligned}$$ ∇ × ∇ × U + λ U = f ( U ) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics.