Blowing-up Behavior for Solutions of Nonlinear Elliptic Equations

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.