A monotonicity result under symmetry and Morse index constraints in the plane

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.