Title

Exponential Decay for Nonlinear Biharmonic Equations

Document Type

Article

Publication Date

2007

Abstract

The purpose of this paper is to establish the exponential decay properties of the solutions for the nonlinear biharmonic equation \[ \hfill \hspace*{9.5pc}\left\{ \begin{array}{l} \Delta^2u + a(x)u = g(x,u), \qquad x\in \mathbb{R}^N, \\[5pt] \displaystyle\lim_{|x|\rightarrow \infty} \ u = 0. \end{array} \right. \hspace*{7pc}\hfill{\hbox{(*)}} \] We introduce the fundamental solutions for the linear biharmonic operator Δ2 - λ if λ < 0. By applying some properties of Hankel functions, which are the solutions of Bessel's equation, we obtain the asymptotic representation of the fundamental solution of Δ2 - λ at ∞ and 0. Asymptotic estimates of the solutions of (*) can be obtained from the properties of the fundamental solutions of Δ2 - λ.

DOI

10.1142/S0219199707002629