One can rarely identify extremizers of nontrivial inequalities, yet one can not infrequently show that extremizers exist. Thus one asks what qualitative and quantitative properties can be established. One way to attack such questions is to exploit Euler-Lagrange equations which extremizers must satisfy. Inequalities involving L^p norms, with p not equal to 2, lead to nonlinear equations. In this talk we discuss the nonlinear, nonlocal Euler-Lagrange equation which arises in connection with such an inequality for the Radon transform. We show that all solutions are infinitely differentiable, and have a certain rate of decay at spatial infinity. (joint work with Qingying Xue)