The Existence Of Homoclinic Orbit For A Class Of Second-order Singular Differential Equation

This thesis mainly concerns with the existence of solutions to a class of singularboundary value problem:(p(t)u' )' = c(t)p(t)f(u), u' (0) = 0, u(+∞) = 0in the half-line [0, +∞), where p(0) = 0. The singular problems appear in the studyof radial solutions of the elliptic equation u = f(u) in a ball or in RN. They also arefrom the research of formation of microscopical bubbles in a nonhomogeneous ?uid.We are interested in strictly decreasing positive solutions of this problem in [0,∞)(homoclinic solution). By analysing the properties of certain solutions associatedwith the singular differential equation, sufficient conditions of the existence of suchsolutions are obtained by applying the critical point theory and by using shootingargument. Recent results in the literature are generalized and significantly improved.In Chapter 2, the existence of strictly decreasing positive solutions is studied fora class of singular boundary value problem by using critical point theory. we firstdevelop a Hilbert space and define a variational functional for the singular boundaryvalue problem. Some useful inequalities and properties are proved. Then applyingthe minimax principle to study the existence of positive solutions for the singularboundary value problem. Due to the fact that the positive solutions minimize the cor-responding variational functional, a strictly decreasing positive solution is obtainedby some analytic techniques. We study in Chapter 3 the existence of positive solu-tions for a class of singular initial value problem by the Banach contraction mappingtheorem, and discuss some basic properties of solutions. In Chapter 4, by using theshooting argument and the results obtained before, several existence criteria of ho-moclinic solution of the singular problem are given in the case that f is bounded orsatisfies sublinear growth condition respectively.