Scalar reaction-diffusion equations on a ball in $R^N$, $N\geq 2$,
with  radially symmetric  nonlinearities  and Dirichlet  boundary
condition are considered.

If the nonlinearlity is nonincreasing in the
radial  variable (in  particular if  it is  independent of  it)  it is
proved that the  stable and unstable manifolds of  any two nonnegative
equilibria intersect  transversally. The crucial property  used in the
proof is that the unstable manifold of a positive equilibrium consists
of  radially symmetric  functions.

In the second part of the paper,  an equation  is constructed
that  admits   two  radially  symmetric   equilibria  whose  invariant
manifolds  intersect nontransversally.

In  the appendix,  examples of spatially  homogeneous equations
with positive  equilibria  with high Morse indices are given.