On Dancer's conjecture for stable solutions with sign-changing nonlinearity

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in ${\mathbb{R}}^n$ with sign-changing nonlinearity $f$. Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of $f$ at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer.