The branch of positive solutions to a semilinear elliptic equation on \(\mathbb{R}^N\)
Summary: For a large class of functions \(f\), we consider the nonlinear elliptic eigenvalue problem \[ \begin{aligned} -\Delta u(x)+f\bigl(x,u(x)\bigr)=\lambda u(x)\quad & \text{for }x \in\mathbb{R}^N,\\ \lim_{|x|\to\infty} u(x)=0, \quad u(x)>0\quad & \text{for all }x\in \mathbb{R}^N.\end{aligned} \] If the lowest eigenvalue of the linearization lies below the essential spectrum, it is known [*, \textit{H. Jeaujean}, \textit{M. Lucia} and \textit{C. A. Stuart}, Math. Z. 230, No. 1, 79-105 (1999)] that a global branch, \({\mathcal C}\), of solutions bifurcates at this value of \(\lambda\). The main results of the present paper give properties of \(f\) which enable us to provide explicit lower bounds for \(\sup\{\lambda\mid(\lambda,u)\in{\mathcal C}\}\). Our approach is based on the construction of supersolutions which enable us to refine the alternative established in [*] concerning the global behaviour of \({\mathcal C}\).
scientific article; zbMATH DE number 1319604