Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Pohozhaev's identity for radial solutions of $p$-Laplace equations

Philip Korman

TL;DR

This work extends the classical Pohozhaev identity to radial solutions of $p$-Laplace equations by deriving a generalized radial identity parameterized by a test function $\psi$ and the nonlinear flux $\varphi(t)=t|t|^{p-2}$. The authors recover the classical identity in the linear limit when $L[\psi]=0$ (e.g., with $\psi=r$ or $\psi=r^{n-1}$) and prove a central integral relation for radial problems, enabling applications such as nonexistence results in Brezis–Nirenberg-type settings. They illustrate the approach with consequences for specific nonlinearities and discuss sharpness and numerical aspects, as well as potential extensions to $f(r,u)$ and to systems. Overall, the paper unifies and extends radial Pohozaev-type identities within the nonlinear $p$-Laplacian framework, with implications for existence and nonexistence results in bounded domains.

Abstract

We derive a generalized Pohozhaev's identity for radial solutions of $p$-Laplace equations, by using the approach in [5], thus extending the work of H. Brézis and L. Nirenberg [2], where this identity was implicitly used for the Laplace equation.

Generalized Pohozhaev's identity for radial solutions of $p$-Laplace equations

TL;DR

This work extends the classical Pohozhaev identity to radial solutions of -Laplace equations by deriving a generalized radial identity parameterized by a test function and the nonlinear flux . The authors recover the classical identity in the linear limit when (e.g., with or ) and prove a central integral relation for radial problems, enabling applications such as nonexistence results in Brezis–Nirenberg-type settings. They illustrate the approach with consequences for specific nonlinearities and discuss sharpness and numerical aspects, as well as potential extensions to and to systems. Overall, the paper unifies and extends radial Pohozaev-type identities within the nonlinear -Laplacian framework, with implications for existence and nonexistence results in bounded domains.

Abstract

We derive a generalized Pohozhaev's identity for radial solutions of -Laplace equations, by using the approach in [5], thus extending the work of H. Brézis and L. Nirenberg [2], where this identity was implicitly used for the Laplace equation.
Paper Structure (3 sections, 3 theorems, 30 equations, 2 figures)

This paper contains 3 sections, 3 theorems, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An application
  3. The $p$-Laplace case

Key Result

Theorem 1.1

Let $u(r) \in C^2[0,1]$ be a solution of and let $\psi(r) \in C^2[0,1]$. Then where $L[\psi]=r^2\psi"-(n-1)r\psi'+(n-1)\psi$.

Figures (2)

  • Figure 1: Solution curve of the Brezis-Nirenberg problem (\ref{['p3a']})
  • Figure 2: Solution curve of the supercritical problem (\ref{['bnir2']})

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 3.1