A semilinear elliptic equation, $-\Delta u=\lambda f(u)$, is studied in a ball with the Dirichlet boundary condition. For a closed subgroup $G$ of the orthogonal group, it is proved that the number of non-radial $G$ invariant solutions diverges to infinity as $\lambda$ tends to $\infty$ if $G$ is not transitive on the unit sphere.
"Non-radial solutions with orthogonal subgroup invariance for semilinear Dirichlet problems." Topol. Methods Nonlinear Anal. 21 (1) 41 - 51, 2003.