Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations
type: Item
instance of: scholarly article
title: Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations
zbMATH Open document ID: 1174.35339
author: Takahiro Hashimoto
published in: Discrete and Continuous Dynamical Systems
publication date: 2004-02-23T00:00:00Z
review text: The author considers the following equation with real parameter \(\alpha\) in bounded domains \[ \begin{gathered} \Delta^2 u= |x|^\alpha|u|^{q-2} u\quad\text{in }\Omega,\\ u= {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(1< q<\infty\) and \(\Omega\subset\mathbb R^N\) \((N> 4)\) is a domain with smooth boundary. He is interested in the existence and nonexistence of nontrivial solutions for (1) depending on \(\alpha\), \(q\) and \(N\). To prove the existence theorem for (1) the author uses a weighted Sobolev-Poincaré type inequality, while the nonexistence results are shown by using a Pokhozhaev-type identity for fourth-order equations.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 2046207
zbMATH Keywords: biharmonic operators
zbMATH Keywords: existence and nonexistence theorems
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2046207