Exact Multiplicity Of Nodal Solutions For Some Two-point Boundary Value Problems Of Second Order Ordinary Differential Equations

Existence and exact multiplicity of solutions for ordinary differential equations can be applied in many fields. We investigate the uniqueness and exact multiplicity of nodal solutions for some two-point boundary value problems of second order ordinary differential equations. A nodal solution of a problem is the one that has only simple zeros. A positive or negative solution is a special nodal solution. This dissertation consists of five chapters.The first chapter is prolegomenon, in which the studying background, theoret-ical frame of this thesis are exhibited. And it displays the main problems we study and the main results we obtain.In the second chapter, applying local bifurcation theory and implicit function theorem, we study the following boundary value problems u"+λf(u)= 0, t∈(0,1), (0.0.1) u'(0)= 0, u(1)= 0, whereλ> 0 is a parameter. Under the condition that uf(u) has negative value, exact multiplicity of nodal solutions is obtained for every givenλ.We pay our attention in the third chapter to the boundary value problems u"|h(t)u'|λf(u)= 0, t∈(0,1), (0.0.2) u(0)= 0, u'(1)= 0, whereλ> 0 is a parameter and h(·) is a weight function. With the condition that f(u)/u is monotonic, the global structure of positive solutions set for (0.0.2) is studied. For any givenλ, we obtain that (0.0.2) has no or exact one positive solution.In addition,we also discuss the boundary value problems u"|λa(t)f(u)= 0, t∈(0,1), (0.0.3) u' (0) = 0, u(1) = 0, whereλ>0 is a parameter and a(·)is a weight function in this chapter.When the monotonicity of f(u)/u changes only once,the global structure of positive solutions set for (0.0.3)is studied.For any givenλ,We obtain that(0.0.2)has no or exact one or exact two positive solutions.The result is brand new since there is no paper before to discuss the exact multiplicity of positive solutions for(0.0.3)under the above conditions.The emphasis in the fourth chapter is to discuss the exact multiplicity of nodal solutions for the boundary value problems u"+λa(t)f(u)=0, t∈(0,1), (0.0.4) u(0)=0, u(1)=0, whereλ>0 is a parameter and a(·)is a weight function.In the first two subsections, we consider the global structure of nodal solutions sets for(0.0.4)when fo,f∞∈(0,∞),or fo,f±∞∈(0,∞)and f+∞≠f-∞.For any givenλ,there is has no or exact one or exact two nodal solutions with specific nodal numbers.In the following two subsections the casc f0=0,f∞=∞(rcsp.f0=∞,f∞=0)is considered. We obtain the exact multiplicity of nodal(resp. positive and negative)solutions whenλ-1.The exact multiplicity of nodal solutions result for the boundary value problems-y"+ρy=λ(y+1)+-α(y+1)-—λ, t∈(0,π), (0.0.5) y'(0)=y'(π)=0 is of our interest in the fifth chapter.The main tools we use in this chapter are the Rabinowitz global bifurcation theorem and Morse index theory.ρ>0 is a constant cnough small,0≤α<ρis also a constant,λ≥ρis a parameter.We determine in this chapter there are exact 2κsolutions of(0.0.5)whenλκ<λ<λκ+1(κ≥1)withλκas theκ-th eigenvalue of the corresponding linear problem of(0.0.5).