Power law convergence and concavity for the Logarithmic Schr\"odinger equation

TL;DR

This paper investigates the concavity properties of solutions to the Logarithmic Schrödinger equation, constructing solutions with concave logarithms through auxiliary problems and advanced mathematical techniques.

Contribution

It introduces a novel method to analyze concavity for solutions with superlinear, sign-changing sources by connecting auxiliary Lane-Emden problems to the logarithmic case.

Findings

Constructed solutions with concave logarithm for the Logarithmic Schrödinger equation.

Developed new Liouville theorems on convex epigraphs applicable to this context.

Provided insights into concavity properties for superlinear, sign-changing source equations.

Abstract

We study concavity properties of positive solutions to the Logarithmic Schr\"odinger equation $-\Delta u=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-\Delta u = \sigma\, (u^q-u)$ and build, for any $\sigma>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $\sigma_q=2/(q-1)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schr\"odinger equation such that $\log u$ is concave. This seems to be one of the few attempts at 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.