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

Abstract

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problems $u^{″}+a(t)f(u)=0$, $t\in (0,1)$, $u(0)=0$, and $u(1)=0$, where $f\in C(ℝ,ℝ)$ satisfies $f(0)=0$ and the limits $f_{\infty }={\text{lim}}_{|s|\to \infty }(f(s)/s)$, $f_0={\text{lim}}_{|s|\to 0}(f(s)/s)\in \{0,\infty \}$. Weight function $a(t)\in C^1[0,1]$ satisfies $a(t)>0$ on $[0,1]$.