Two-Point Boundary Value Problems For Second Order Singular Differential Equations

With the development of science and technology, there appears all kinds of nonlinear problems in the fields of physics, chemistry, biology, economics, engineering etc. Numerous nonlinear problems involving gas dynamics, fluid mechanics, boundary layer theory etc, can be described by ordinary differential equations with singularity. In view of then w(?)d(?) application and profound (?)athematical meanings, the study of singular ordinary differential equations attracted a great attention. Many results concerning the boundary value problems of ordinary differential equations with singularity are obtained. In recent years, the classic Monge-Amp(?)re equation, which is one type of important nonlinear ordinary differential equations, has increasing significance in the areas of applied mathematics. Therefore, the study of the existence of positive solutions for this type of equations has become a hot topic for many mathematical scholars.In this thesis, we mainly consider the boundary value problems as followsShouchuan (?) est(?)blished the sufficie(?) conditi(?)ns for the existe(?)e of at (?)ost (?)ne positive or two positive solutions for problem (â… ), where the function f may be singular at v=0, f is continuous and increasing on [0,+∞). Lions consider the boundary value problems as followswhere B={x∈Rⁿ:|x|<1} is the unit ball in Rⁿ and D²u=(?) is the Hessianof u. As noticed by Lions, the particular function f(u)=uⁿ acts like a "linear" term to the fully nonlinear operation det(D²u). In fact, Lions proved the existence of a unique eigenvalueλ₁ for the boundary value problem (â…¡) with f(u)=uⁿ.Specifically,λ₁>0 and the corresponding eigenfunction (?)₁ is negative convex, and any other eigenfunction would be a positive constant multiple of (?)₁. Furthermore,λ₁ acts like a bifurcation point for the boundary value problem (â…¡) with general functions /, which reflects the wellknown properties of the first eigenvalues of linear second-order elliptic operators or more generally of positive operators as given by the famous Krein-Rutman theorem. For this reason, the so-called sublinear or superlinear function f(u), is defined in relation to uⁿ. Kutev obtained the existence of a unique nontrivial convex radially symmetric solution of the boundary value problem (â…¡) with f(u)=u^p,(?)_p,0