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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 $[0,\infty)$. Here $B<0$ is a parameter, $p(0)=0$ and $p'>0$ on $(0,\infty),$ $f(L)=0$ for some $L>0$ and $xf(x)<0$ if $x<L$, $x\neq 0$. By means of this result, the existence of a strictly increasing solution of this problem satisfying $u(\infty)=L$ 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 $L$.

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