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Subhomogeneous and singular quasilinear Emden-type ODE. (English) Zbl 1058.34505
The author studies the problem \[ (r^a| u'|^{p-2}u')'+ r^af(r)u^\gamma_+= 0, \;r> 0;\quad u(0)= \alpha,\;u'(0)= 0, (\text{E})_\alpha \] where \({}'= d/dr\), \(a> 1\), \(1< p< a+1\), \(0\leq\gamma< p-1\), \(f\) is bounded, continuous and strictly positive in \([0,+\infty)\) with \(f(r)\sim r^{-\theta}\) as \(r\to+\infty\), for some \(\theta> p\). He proves that (E)\(_\alpha\) has a unique solution \(u_\alpha\) and that there exists some \(\alpha^*\) such that \(u_\alpha\) is strictly positive for \(\alpha> \alpha^*\), while \(u_\alpha\) has a positive zero for \(0<\alpha<\alpha^*\).
MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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