We construct a gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.