Bubble-type solutions of nonlinear singular problems
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, infinity). Here B   0 on (0, infinity), f(L) = 0 for some L > 0 and xf(x) < 0 if x < L, x not equal 0. By means of this result, the existence of a strictly increasing solution of this problem satisfying u(infinity) = 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. (C) 2009 Elsevier Ltd. All rights reserved.
homoclinic solution singular ordinary differential equation of the second order time singularities unbounded domain