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.
Categories: math.AP