Exact Multiplicity of Sign-Changing Solutions for a Class of Second-Order Dirichlet Boundary Value Problem with Weight Function

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problems u '' + a(t)f(u) = 0, t epsilon (0, 1), u(0) = 0, and u(1) = 0, where f epsilon C(R,R) satisfies f(0) = 0 and the limits f(infinity) = lim(vertical bar s vertical bar ->infinity)(f(s)/s), f(0) = lim(vertical bar s vertical bar -> 0)(f (s)/s) epsilon {0, infinity}. Weight function a(t) epsilon C-1[0, 1] satisfies a(t) > 0 on [0, 1].