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We are interested in the existence of normalized solutions to the problem (- 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 -ball. Moreover, we find also a solution to the related curl–curl problem ∇ × ∇ × 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. [ABSTRACT FROM AUTHOR]

We develop a framework shedding light on common features of coupled variational formulations arising in electromagnetic scattering and acoustics. We show that spurious resonances haunting coupled domain-boundary formulations based on direct boundary integral equations of the first kind originate from the formal structure of their Calderón identities. Using this observation, the kernel of the coupled problem is characterized explicitly, and we show that it completely vanishes under the exterior representation formula. [ABSTRACT FROM AUTHOR]