Power law convergence and concavity for the logarithmic Schrödinger equation

Abstract We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $$-\Delta u=u\, \log u^2$$ - Δ u = u log u 2 in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane–Emden problems $$-\Delta u = \sigma \, (u^q-u)$$ - Δ u = σ ( u q - u ) and build, for any $$\sigma >0$$ σ > 0 and $$q>1$$ q > 1 , solutions $$u_q$$ u q such that $$u_q^{(1-q)/2}$$ u q ( 1 - q ) / 2 is convex. By choosing $$\sigma _q=2/({q-1})$$ σ q = 2 / ( q - 1 ) and letting $$q \rightarrow 1^+$$ q → 1 + we eventually construct a solution u of the Logarithmic Schrödinger equation such that $$\log u$$ log u 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.