Spectrum and global bifurcation results for nonlinear second-order problem on all of \(\mathbb{R}\) (Q2105335)

In this paper, the authors consider the following second order problem \[ \left\{ \begin{array}{l} -u''(t)+a(t)u(t)=\lambda b(t)f(u(t)),\ t\in\mathbb{R}\\ u(t)\longrightarrow 0\ as\ \left|t\right|\longrightarrow\infty \end{array}\right.\tag{1} \] where \(\lambda>0\) is a parameter, \(a,b\in C(\mathbb{R},]0,\infty[)\) such that \(\lim_{\left|t\right|\longrightarrow\infty}\frac{a(t)}{b(t)}=0\), \(f:\mathbb{R}\longrightarrow\mathbb{R}\) is a continuous function with \(sf(s)>0\) for \(s\neq 0\). For the linear case, i.e., \(f(u)=u\), they investigate the existence of principal eigenvalue of \((1)\). For the nonlinear case, depending on the behavior of \(f\) near \(0\) and \(\infty\), they obtain asymptotic behavior and the existence of homoclinic solutions of \((1)\). The proof of these results is based upon bifurcation technique.