Stable solutions on $\R^n$ and the primary branch of some non-self-adjoint convex problems

Abstract

We prove that if $n = 2$ or $3$, the problem $- \Delta u = e^u$ on $R^n$ has no stable negative solution. We then use this to remove self-adjointness conditions in a paper of Crandall and Rabinowitz on the primary branch of positive solutions of a nonlinear boundary-value problem.