  This paper deals with solutions of semilinear elliptic equations of the type
\[ \left\{\begin{array}{ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega,
\\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where $\Omega$ is
a radially symmetric domain of the plane that can be bounded or unbounded. We
consider solutions $u$ that are invariant by rotations of a certain angle
$\theta$ and which have a bound on their Morse index in spaces of functions
invariant by these rotations. We can prove that or $u$ is radial, or, else,
there exists a direction $e\in \mathcal S$ such that $u$ is symmetric with
respect to $e$ and it is strictly monotone in the angular variable in a sector
of angle $\frac{\theta}2$. The result applies to least-energy and nodal
least-energy solutions in spaces of functions invariant by rotations and
produces multiplicity results.

Analysis of PDEs