Calculus of Variations and Geometric Measure Theory

N. Almousa - J. Assettini - M. Gallo - M. Squassina

Concavity properties for quasilinear equations and optimality remarks

created by gallo on 31 May 2023
modified on 15 May 2024


Published Paper

Inserted: 31 may 2023
Last Updated: 15 may 2024

Journal: Differential Integral Equations
Volume: 37
Number: 1-2
Pages: 1-26
Year: 2023
Doi: 10.57262/die037-0102-1

ArXiv: 2305.09982 PDF


In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schr\"odinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2}
\nabla u
2 = f(u) \quad \hbox{in $\Omega$},$$ where $\Omega$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $\varphi:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\varphi(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.