*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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