Power law convergence and concavity for the Logarithmic Schrödinger equation

Abstract

We study concavity properties of positive solutions to the Logarithmic Schrödinger equation $-Δu=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-Δu = σ\, (u^q-u)$ and build, for any $σ>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $σ_q=2/(1-q)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schrödinger 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.

Figures (2)

• Figure 1: Graphs of $u$ and $\sqrt{u}$, where $-\Delta u = u \log u^2$ in $B_2(0) \subset {\mathbb R}^2$.
• Figure 2: Graph of $-\sqrt{-\log(u/\|u\|_{\infty})}$, where $-\Delta u = u \log u^2$ in $B_2(0) \subset {\mathbb R}^2$.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Liouville theorem on convex epigraphs
• Theorem 1.5
• Remark 1.6: On Theorem \ref{['thm_main_log']}
• Remark 1.7: On Theorem \ref{['thm_main_power']}
• Remark 1.8: On Theorem \ref{['thm_convergence']}
• Remark 1.9: On Theorem \ref{['thm_main_liouville']}
• Remark 1.10: On Theorem \ref{['thm_diff_sign']}