Structural change of solutions for a scalar curvature equation

Abstract

A semilinear elliptic equation \[ \Delta u + \{1 + {\varepsilon} k(|x|) \} u^p=0, \quad x \in {\bf R}^n, \] is studied, where $n>2$ and ${\varepsilon}$ is a small parameter. It is known that for $p=(n+2)/(n-2)$ fixed, the structure of radial solutions drastically changes under the perturbation ${\varepsilon} k(|x|)$. In this paper it is shown that such a structural change can be understood in a natural way if the exponent $p$ also is taken as a parameter. The Pohozaev identity plays an important role in the perturbation analysis.