In a joint work with Xavier Cabre, we study the instability properties of some saddle solutions of the semilinear elliptic equations $-\Delta u=u-u^3$ in the whole $\R^{2m}$.

In 1995 M. Schatzman proved that in dimension $2$ there exists a saddle solution and it is unstable.

We prove that saddle solutions exist for every even dimension and that such solutions are unstable in dimension $4$. Furthermore, we establish that every solution vanishing on the Simons cone is unstable in dimension $4$.