We consider the semilinear elliptic equation
$$
\Delta u+f(u)=0, \text{ on }\mathbb R^N,
$$
where  $f$ is of class $C^1$ and satisfies the conditions  $f(0)=0$, $f'(0)<0$. By a ground state of this equation we mean  a  positive solution that decays to zero at infinity. Any such solution is necessarily radially symmetric about some point. If $N=1$, the ground state, if it exists, is always unique (up to a shift in $x$), nondegenerate and has Morse index equal to one. We show that none of these statements is valid in general if $N\ge 2$.