## Spinning Q-balls for the Klein-Gordon-Maxwell equations.(English)Zbl 1194.78009

The paper deals with the nonlinear Klein-Gordon-Maxwell equations $$\frac{\partial^2\psi}{\partial t^2} - \nabla\psi +W'(\psi)=0$$, where the nonlinear term $$W$$ is assumed to be positive and $$W(0)=0$$. The vortices in the nonlinear Klein-Gordon equation are often considered in the Physics literature with the name of spinning $$Q$$-balls. The main result of the paper is the proof of existence of spinning $$Q$$-balls for the nonlinear Klein-Gordon-Maxwell equations, that is, three-dimensional vortex-solutions which are finite energy, stationary solutions such that the matter field has a nontrivial angular momentum and the magnetic field looks like the field created by a finite solenoid.

### MSC:

 78A25 Electromagnetic theory (general) 47J30 Variational methods involving nonlinear operators 35J50 Variational methods for elliptic systems 81V10 Electromagnetic interaction; quantum electrodynamics

### Keywords:

solitary wave; votrex; spinning Q-balls; Klein-Gordon equation
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### References:

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