Calculus of Variations and Geometric Measure Theory

M. Gallo - S. J. N. Mosconi - M. Squassina

Power law convergence and concavity for the logarithmic Schrödinger equation

created by squassina on 31 Oct 2024

[BibTeX]

Preprint

Inserted: 31 oct 2024
Last Updated: 31 oct 2024

Pages: 53
Year: 2024

Abstract:

We study concavity properties of positive solutions to the logarithmic Schrödinger equation $-\Delta u=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary 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 $\sigma_q=2/(1-q)$ 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 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.


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