Solitons in several space dimensions: Derrick's problem and infinitely many solutions

In this paper we study a class of Lorentz invariant nonlinear field equations in several space dimensions. The main purpose is to obtain soliton-like solutions. These equations were essentially proposed by C. H. DERRICK in a celebrated paper in 1964 as a model for elementary particles. However, an existence theory was not developed. The fields are characterized by a topological invariant, the charge. We prove the existence of a static solution which minimizes the energy among the configurations with nontrivial charge. Moreover, under some symmetry assumptions, we prove the existence of infinitely many solutions, which are constrained minima of the energy. More precisely, for every $n\in\mathbb{N}$ there exists a solution of charge $n$.