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Nonlinear Schrödinger equations with strongly singular potentials. (English) Zbl 1197.35263

Summary: We look for standing waves for nonlinear Schrödinger equations
\[ i\frac{\partial\psi}{\partial t}+ \Delta\psi- g(|y|)\psi- W'(|\psi|) \frac{\psi}{|\psi|}= 0 \]
with cylindrically symmetric potentials \(g\) vanishing at infinity and non-increasing, and a \(C^1\) nonlinear term satisfying weak assumptions. In particular, we show the existence of standing waves with non-vanishing angular momentum with prescribed \(L^2\) norm. The solutions are obtained via a minimization argument, and the proof is given for an abstract functional which presents a lack of compactness. As a specific case, we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35A15 Variational methods applied to PDEs
47N20 Applications of operator theory to differential and integral equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V45 Atomic physics
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