Power law convergence and concavity for the logarithmic Schrödinger equation

We study concavity properties of positive solutions to the Logarithmic Schrödinger equation -Δu=ulogu2 in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane–Emden problems -Δu=σ(uq-u) and build, for any σ>0 and q>1, solutions uq such that uq(1-q)/2 is convex. By choosing σq=2/(q-1) and letting q→1+ we eventually construct a solution u of the Logarithmic Schrödinger equation such that logu is concave. This seems one of the few attempts in studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.