Existence and nonexistence of solutions of second-order nonlinear boundary value problems

We study the nonlinear boundary value problem consisting of the equation −y″+q(t)y=w(t)f(y) on [a,b]−y″+q(t)y=w(t)f(y) on [a,b] and a general separated homogeneous linear boundary condition. By comparing this problem with a corresponding linear Sturm–Liouville problem we obtain conditions for the existence and nonexistence of solutions of this problem. More specifically, let λn,n=0,1,2,…λn,n=0,1,2,…, be the nn-th eigenvalues of the corresponding linear Sturm–Liouville problem. Then under certain assumptions, the boundary value problem has a solution with exactly nn zeros in (a,b)(a,b) if λnλn is in the interior of the range of f(y)/y,y∈(0,∞)f(y)/y,y∈(0,∞); and does not have any solution with exactly nn zeros in (a,b)(a,b) if λnλn is outside of the range of f(y)/y,y∈(0,∞)f(y)/y,y∈(0,∞). These conditions become necessary and sufficient when f(y)/yf(y)/y is monotone. The existences of multiple and even an infinite number of solutions are derived as consequences. We also discuss the changes of the number and the types of nontrivial solutions as the interval [a,b][a,b] shrinks, as qq increases in a given direction, and as the boundary condition changes.