Singular Nonlinear Boundary Value Problem for the Bubble Type or Droplet Type Solutions in Nonlinear Physics Models

For a second order nonlinear ordinary differential equation (ODE), a singular boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble type (droplet type) solutions are sought. Being defined on a semi infinite interval 0 < r <∞, this ODE, with a polynomial nonlinearity of the third order with respect to a desired function, possesses a regular singular point as r → 0 and an irregular one as r →∞. Using some results for singular Cauchy problems (CPs) and stable initial manifolds (SIMs), we give the restrictions to the parameters for correct mathematical statement of the above singular nonlinear BVP, solving as well an accompanying problem concerning the transfer of the boundary condition from a singular point into a close regular one. Due to a certain variational approach and some results for so called ground state problem, the necessary and sufficient conditions for existence of bubble type or droplet type solutions are discussed (in the form of additional restrictions to the parameters) and some estimates are obtained. For a bubble model in the modern theory of nonhomogeneous or two phase fluids with the equations of state depending on the derivatives, the singular nonlinear BVP under consideration has been posed and partially studied in [1] including numerical simulation of the problem (some preliminary results have been announced also in [2] [4]). In the present work we give briefly some results concerning a more complete and accurate theoretical analysis of this BVP and its applications. We don't present here the numerical methods and computational results. The detailed analytical numerical investigation of the above singular nonlinear BVP, including physical interpretation of the numerical results, is assumed to be published in [5].