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

Bellazzini, Jacopo; Bonanno, Claudio

Proc. R. Soc. Edinb., Sect. A, Math. 140, No. 4, 707-721 (2010).

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

Keywords:

nonlinear Schrödinger equations; standing waves; minimization arguments; hydrogen atom equation