Seminari del Dipartimento di Matematica

Seminario del 2025

Concavity properties for solutions of the Logarithmic Schrödinger equation

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

I will discuss the existence of a solution to the logarithmic Schrödinger equation in a bounded convex domain, with the property that log u is concave. Since the reaction is sign-changing and non-monotone, the classical techniques to attack the problem fail. We instead rely on a continuity argument for the approximating Lane-Emden problems based on the heuristic argument. We will discuss the optimality of the result, exhibiting for any α > 0 a solution such that u^a is not concave. This is a joint work with M. Squassina and M. Gallo.