A note on nonlinear biharmonic equations with negative exponents
type: Item
instance of: scholarly article
title: A note on nonlinear biharmonic equations with negative exponents
zbMATH Open document ID: 1262.35098
author: Ignacio Guerra
published in: Journal of Differential Equations
publication date: 2012-10-26T00:00:00Z
review text: This paper is concerned with the asymptotic behavior of positive radially symmetric solutions to the equation \(\Delta^2 u=-u^{-q}\) in \(R^2\). The following are the main results of the paper. Theorem 1. (a) For \(q>3\) and given any \(\alpha>0\) there exists a unique radially symmetric solution such that \[ \lim_{|x|\rightarrow \infty}\frac{u(x)}{|x|}=\alpha. \] (b) For \(q=3\) there exists a unique radially symmetric solution such that \[ \lim_{|x|\rightarrow \infty}\frac{u(x)}{|x|\log|x|^{1/4}}=2^{1/4}. \] (c) For \(1<q<3\) there exists a unique radially symmetric solution such that \[ \lim_{|x|\rightarrow \infty}\frac{u(x)}{|x|^{4/(q+1)}}=K^{-1/(q+1)}, \quad\text{ where }K=\tau(2-\tau)(\tau+1)(\tau-1),\quad \tau=4/(q+1). \] Theorem 2. Let \[ \alpha=\frac{1}{8\pi}\int_{R^3}u^{-q}(x)dx. \] (a) If \(q=4\) then \[ \lim_{|x|\rightarrow\infty}\frac{u(x)-\alpha|x|}{\log|x|}=-\frac{1}{2\alpha^4}. \] (b) If \(3<q<4\) then \[ \lim_{|x|\rightarrow\infty}|x|^{q-4}u(x) =\frac{1}{2\alpha^q}\Big(\frac{1}{3-q}-\frac{1}{4-q}+\frac{1}{3(5-q)}-\frac{1}{3(2-q)} \Big). \]
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B06
zbMATH DE Number: 6099370
zbMATH Keywords: nonlinear biharmonic equation
zbMATH Keywords: negative exponent
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: radially symmetric solution
MaRDI profile type: Publication
full work available at URL: j.jde.2012.08.037
OpenAlex ID: W1982372883
cites work: Applications of centre manifold theory
cites work: Nonlinear biharmonic equations with negative exponents
cites work: Multiplicity of solutions for a fourth order equation with power-type nonlinearity
cites work: Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in \(\mathbb{R}^3\)
cites work: Asymptotic behaviour of ground states
cites work: Q4444772
cites work: Introduction to Applied Nonlinear Dynamical Systems and Chaos
cites work: Exact solutions of nonlinear conformally invariant integral equations in \(\mathbf R^3\)
cites work: Uniqueness theorem for integral equations and its application
DOI: 10.1016/J.JDE.2012.08.037
description: scientific article; zbMATH DE number 6099370