Concavity properties for quasilinear equations and optimality remarks

In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schrödinger equations of the type $$-\mathrm{d}\mathrm{i}\mathrm{v}(a(u)\mathrm{\nabla }u)+\frac{a^{\mathrm{\prime }}(u)}{2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2=f(u)\text{in }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is a convex bounded domain of ${\mathbb{R}}^N$. In particular, we search for a function $\phi :\mathbb{R}\to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\phi (u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.