## Mathematics and Statistics Faculty Publications

#### Title

Exponential Decay for Nonlinear Biharmonic Equations

Article

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

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