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The Classification of all weak solutions to $-Δu={u^{-γ}}$ in the half-space

Luigi Montoro, Luigi Muglia, Berardino Sciunzi

TL;DR

We address the problem of classifying positive weak solutions to $- abla u=\frac{1}{u^\gamma}$ in the half-space $\mathbb{R}^N_+$ with $u=0$ on the boundary. The authors develop a three-part strategy: establish local boundedness via a Serrin-Moser iteration adapted to the singular nonlinearity, extend regularity up to the boundary by barrier arguments, and prove 1-D symmetry using a Kelvin transform, reducing the problem to a boundary-value ODE. The main result proves a sharp dichotomy: for $\gamma>1$ all solutions are one-dimensional and either have a power-type profile or a scaled ODE profile; for $0<\gamma\le1$ no global positive solutions exist. This completes the classification for semilinear elliptic equations with power nonlinearities in a half-space and provides explicit profiles and nonexistence results that can impact boundary-layer and MEMS-type models.

Abstract

We provide the classification of all the positive solutions to $-Δu=\frac{1}{u^γ}$ in the half space, under minimal assumption.

The Classification of all weak solutions to $-Δu={u^{-γ}}$ in the half-space

TL;DR

We address the problem of classifying positive weak solutions to in the half-space with on the boundary. The authors develop a three-part strategy: establish local boundedness via a Serrin-Moser iteration adapted to the singular nonlinearity, extend regularity up to the boundary by barrier arguments, and prove 1-D symmetry using a Kelvin transform, reducing the problem to a boundary-value ODE. The main result proves a sharp dichotomy: for all solutions are one-dimensional and either have a power-type profile or a scaled ODE profile; for no global positive solutions exist. This completes the classification for semilinear elliptic equations with power nonlinearities in a half-space and provides explicit profiles and nonexistence results that can impact boundary-layer and MEMS-type models.

Abstract

We provide the classification of all the positive solutions to in the half space, under minimal assumption.
Paper Structure (3 sections, 4 theorems, 54 equations)

This paper contains 3 sections, 4 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Regularity of the solutions
  3. 1-D symmetry

Key Result

Theorem 2

Let $u$ be a solution to MP. Then, in the case $\gamma>1$, all the solutions $u$ are one dimensional, i.e. and, either or where $v(t)\in C^2(\mathbb{R}_+)\cap C(\overline{\mathbb{R}_+})$ is the unique positive solution to In the case where $0<\gamma\leq 1$, there are no solutions to MP.

Theorems & Definitions (9)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • proof
  • Theorem 5: DPGUI
  • Theorem 6
  • proof
  • proof : Proof of Theorem \ref{['mainthm']}