Advanced Studies in Pure Mathematics

Blowing-up Behavior for Solutions of Nonlinear Elliptic Equations

Tatsuo Itoh

Full-text: Open access

Abstract

We consider the following nonlinear elliptic equations with real parameter $\lambda$: \[ \tag{$P_\lambda$} \Delta u+f(u,\lambda)=0, \quad u>0 \text{ in } \Omega; \quad u=0 \quad \text{on } \partial\Omega, \] where $\Omega$ is a smooth bounded domain in $R^n$ ($n \ge 2$) and $f \ge 0$ satisfies an inequality: \[ f(u,\lambda) \le c_1 + c_2 u^p \quad (c_1,c_2 > 0, p > 1\text{ constants}). \]

We suppose the existence of a family of solutions $\{(u_s, \lambda_s)\}_{0<s\le1}$ of $(P_\lambda)$ with the following properties: $(u_s, \lambda_s) \in C (\bar{\Omega}) \times R$ is continuous in $s$, $\lambda_s$ is bounded, and $\max u_s \to \infty$ ($s \downarrow 0$).

We investigate the asymptotic behavior of solutions near blowing-up points.

Article information

Source
Spectral and Scattering Theory and Applications, K. Yajima, ed. (Tokyo: Mathematical Society of Japan, 1994), 177-186

Dates
Received: 8 February 1993
First available in Project Euclid: 15 August 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1534359750

Digital Object Identifier
doi:10.2969/aspm/02310177

Mathematical Reviews number (MathSciNet)
MR1275401

Zentralblatt MATH identifier
0806.35037

Citation

Itoh, Tatsuo. Blowing-up Behavior for Solutions of Nonlinear Elliptic Equations. Spectral and Scattering Theory and Applications, 177--186, Mathematical Society of Japan, Tokyo, Japan, 1994. doi:10.2969/aspm/02310177. https://projecteuclid.org/euclid.aspm/1534359750


Export citation