Concavity properties for quasilinear equations and optimality remarks

In this paper, we study quasiconcavity properties of solu-tions of Dirichlet problems related to modified nonlinear Schro center dot dinger equations of the type -div(a(u)Vu) + a '(u) 2 |Vu|2 = f(u) in 52, where 52 is a convex bounded domain of RN. In particular, we search for a function phi : R R, modeled on f E C1 and a E C1, which makes phi(u) concave. Moreover, we discuss the optimality of the conditions assumed on the source.