On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p and a parameter \lambda > 0. For both equations we consider Dirichlet boundary conditions in the unit ball B \subset R^n. Regularity of solutions strictly depends on the power p and the parameter \lambda. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.