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 pp and a parameter λ>0λ>0. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂RnB⊂Rn. Regularity of solutions strictly depends on the power pp and the parameter λλ. 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.