A note on the complete rotational invariance of biradial solutions to semilinear elliptic equations

We investigate symmetry properties of solutions to equations of the form $$ -\Delta u = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact completely radial. A similar result holds for the semilinear Laplace-Beltrami equations on the sphere. Furthermore, we show that the condition on the Morse index is sharp. Finally we apply the result in order to estimate best constants of Sobolev type inequalities with different symmetry constraints.