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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 $\Omega$ isa radially symmetric domain of the plane that can be bounded or unbounded. Weconsider solutions $u$ that are invariant by rotations of a certain angle$\theta$ and which have a bound on their Morse index in spaces of functionsinvariant 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 withrespect to $e$ and it is strictly monotone in the angular variable in a sectorof angle $\frac{\theta}2$. The result applies to least-energy and nodalleast-energy solutions in spaces of functions invariant by rotations andproduces multiplicity results.

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