The boundary value problem under consideration is \[ -u''(t)=|u(t)|^{p-1}u(t)-\lambda u(t), t\in (0,1); u(0)=u(1)=0, \tag \(*\) \] where \(p>1\) and \(\lambda\in \mathbb{R}\) is an eigenvalue parameter. As it is well known by \textit{H. Berestycki} [J. Funct. Anal. 40, 1-29 (1981; Zbl 0452.35038)], \((-(n \pi)^2, 0)\in \mathbb{R}\times C^2[0,1]\) are the bifurcation points of \((*)\) for \(n\in \mathbb{N}\). From each of this points, a continuum of nontrivial solutions \((\lambda, u)\) to \((*)\) emanates. In fact, this continuum is a \(C^1\)-curve in \(\mathbb{R}\times C^2[0,1]\). Thus, the study of the nonlinear eigenvalue problem \((*)\) is equivalent to the study of the global behaviour of these bifurcation branches in \(L^2\). Precise asymptotic formulas describing the dependence of \(\lambda\) on \(\|u\|_{L^2(0,1)}\) are developed in the paper (several cases are distinguished, e.g., \(\|u\|_{L^2(0,1)}\to\infty\), \(\|u\|_{L^2(0,1)}\to 0\) and various conditions on \(p\) are superimposed in each case).