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

TL;DR

This paper establishes the existence of normalized solutions for singular polyharmonic equations and curl-curl problems in the critical mass regime, using variational methods on the $L^2$-ball, with applications to nonlinear optics.

Contribution

It introduces novel variational techniques for finding normalized solutions in critical regimes and applies them to complex polyharmonic and curl-curl problems.

Findings

Existence of solutions in the critical mass regime.

Application of variational methods on the $L^2$-ball.

Solutions relevant to nonlinear optics and Maxwell equations.

Abstract

We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-\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{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}+\lambda\mathbf{U}=f(\mathbf{U}), \quad x \in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|\mathbf{U}|^2\,dx=\rho, \end{cases} \end{equation*} which arises from the system of Maxwell equations and is of great importance in nonlinear optics.