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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

${\left(p\left(t\right){u}^{\text{'}}\right)}^{\text{'}}=p\left(t\right)f\left(u\right),\phantom{\rule{3.33333pt}{0ex}}u\left(0\right)=B,\phantom{\rule{3.33333pt}{0ex}}{u}^{\text{'}}\left(0\right)=0$

on the half-line $\left[0,\infty \right)$. Here $B<0$ is a parameter, $p\left(0\right)=0$ and ${p}^{\text{'}}>0$ on $\left(0,\infty \right),$ $f\left(L\right)=0$ for some $L>0$ and $xf\left(x\right)<0$ if $x, $x\ne 0$. By means of this result, the existence of a strictly increasing solution of this problem satisfying $u\left(\infty \right)=L$ 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:
 34B40 Boundary value problems for ODE on infinite intervals 34B16 Singular nonlinear boundary value problems for ODE 76N10 Compressible fluids, general
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