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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