On the stability of the positive radial steady states for a semilinear Cauchy problem involving critical exponents (Q1003896)

This paper is devoted to the study of the Cauchy problem \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta u+K(|x|)u^p \quad \text{ in } {\mathbb{R}}^n\times (0,T), \\ u(x,0)=\varphi(x) \quad \text{ in } {\mathbb{R}}^n, \end{cases} \] where \(n\geq 3, T>0, K(|x|)\) is a local Hölder continuous function in \({\mathbb{R}}^n\setminus \{0\}\), and \(\varphi\not\equiv 0\) is a bounded nonnegative continuous function in \({\mathbb{R}}^n\). Only the radial steady states of the equation are considered. This leads to the initial value problem \[ \begin{cases} u''+\frac {n-1}{r}u'+K(r)u^p=0 \, \text{ for }r>0,\\ u(0)=\alpha>0. \end{cases}. \] The main result of the paper is the proof of the stability of any positive radial steady state when \(K(r)\) satisfies appropriate conditions and \(p\) equals some critical exponent \(p_c\). The methods include the construction of super- and sub-solutions and some estimates of the radial steady states.