A basic problem in nonlinear elasticity is to be able to construct realistic assumptions on the stored energy function which, on the one hand, lead to the existence of solutions to the equilibrium equations and which, on the other hand, enable one to understand the nature of possible irregularities of such solutions. In the context of nonlinear energy functionals, there are very few known results on existence and regularity of weak solutions to the corresponding equilibrium equations. In this presentation, we will examine a recent result on the question of existence of weak equilibrium solutions in nonlinear radial elasticity. Then we will discuss a new result on and approach to the question of regularity of such solutions that replaces the delicate phase plane analysis.