Positivity for the linearized problem for semilinear equations

Abstract

Using recent results of M. Tang [Uniqueness of positive radial solutions for $\Delta u − u + u^p = 0$ on an annulus, J. Differential Equations 189 (2003), no. 1, 148–160], we provide a simple approach to proving positivity for the linearized problem of semilinear equations, which is crucial for establishment of exact multiplicity results, and for symmetry breaking.