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Bubble-type solutions of nonlinear singular problems. (English) Zbl 1190.34029

Summary: The paper describes the set of all solutions of the singular initial problems

(p(t)u ' ) ' =p(t)f(u),u(0)=B,u ' (0)=0

on the half-line [0,). Here B<0 is a parameter, p(0)=0 and p ' >0 on (0,), f(L)=0 for some L>0 and xf(x)<0 if x<L, x0. By means of this result, the existence of a strictly increasing solution of this problem satisfying u()=L is proved under some additional assumptions. In particular cases,this homoclinic solution determines an increasing mass density in centrally symmetric gas bubbles which are surrounded by an external liquid with density L.


MSC:
34B40Boundary value problems for ODE on infinite intervals
34B16Singular nonlinear boundary value problems for ODE
76N10Compressible fluids, general
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