Dancer's stable solution conjecture

Consider the semilinear elliptic equation

$$-\Delta u=f(u)\quad\text{in }\mathbb{R}^{n},$$

where $f$ is a $C^1$ function, and let $u$ be a bounded stable $C^2$ solution, meaning that

$$\int_{\mathbb{R}^{n}}\left(|\nabla\psi|^{2}-f'(u)\psi^{2}\right)\,dx\geq 0$$

for every $\psi\in C_0^\infty(\mathbb{R}^{n})$. A solution is one dimensional if it depends only on one direction. Dancer's stable solution conjecture. If $n\leq 8$, then $u$ is one dimensional. This conjecture concerns the classification of bounded stable solutions for semilinear elliptic equations and was proposed by Dancer. Its status is not established by the supplied text.