Asymptotic behavior of radial solutions for a class of semilinear elliptic equations (Q678061)

The authors are concerned with the asymptotic behavior of the oscillatory radial solutions of a semilinear equation of the form \[ \Delta u+f(|x|,u)=0\quad\text{in }\mathbb{R}^n\quad (n\geq 3), \] and study the related singular initial value problem: \[ u''+{n-1\over r} u'+ (br^\mu|u|^{q-1}+r^\nu|u|^{p-1})u= 0,\quad u(0)=a\quad\text{and} \quad u'(0)=0,\tag{P} \] where \(a\neq 0\), \(b>0\), \(n\geq 3\), \(\mu>-2\), \(\nu>-2\), \(1<q<(n+ 2+2\mu)/(n- 2)\) and \(p=(n+2+2\nu)/(n- 2)= 1+(4+ 2\nu)/(n- 2)>1\). They prove that every solution of (P) oscillates about zero infinitely many times and derive certain asymptotic estimates for the amplitudes and the periods of the oscillations.