Analytical–numerical investigation of bubble-type solutions of nonlinear singular problems

In this work we are concerned about a singular boundary value problem for a second-order nonlinear ordinary differential equation, arising in hydrodynamics and nonlinear field theory, when centrally symmetric bubble-type solutions are sought. We are interested on solutions of this equation which are strictly increasing on the positive semi-axis and have finite limits at zero and infinity. Necessary conditions for the existence of such solutions are obtained in the form of a restriction on the equation parameters. The asymptotic behavior of certain solutions of this equation is analyzed near the two singularities (when r → 0 + and r → ∞ ), where the considered boundary conditions define one-parameter families of solutions. Based on the analytic study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.