Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

An exact multiplicity result of positive solutions for the boundary value problems u '' + lambda a(t)f(u) = 0, t epsilon (0, 1), u`(0) = 0, u(1) = 0 is achieved, where lambda is a positive parameter. Here the function f : [0, infinity) -> [0, infinity) is C(2) and satisfies f(0) = f(s) = 0, f(u) > 0 for u epsilon (0, infinity) for some s epsilon (0, infinity). Moreover, f is asymptotically linear and f '' can change sign only once. The weight function a : [0, 1] -> (0, infinity) is C(2) and satisfies a`(t) < 0, 3(a`(t))(2) < 2a (t)a ''(t) for t epsilon [0,1]. Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for lambda lying in various intervals in R. Moreover, we indicate how to extend the result to the general case.