Symmetry of solutions to semilinear elliptic equations via Morse index

In this paper we prove symmetry results for solutions of semilinear elliptic equations in a ball or in an annulus in $\mathbb{R}^N$, $N \ge 2$, in the case where the nonlinearity has a convex first derivative. More precisely we prove that solutions having Morse index $j \le N$ are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce, under some additional hypotheses on the nonlinearity, that the nodal set of sign changing solutions with Morse index $j \le N$ intersects the boundary of the domain.