  We consider positive solutions of a semilinear Dirichlet problem \[ \Delta
u+\lambda f(u)=0, \;\; \mbox{for $|x|<1$}, \;\; u=0 , \;\; \mbox{when $|x|=1$}
\] on a unit ball in $R^n$. For four classes of self-similar equations it is
possible to parameterize the entire (global) solution curve through the
solution of a single initial value problem. This allows us to derive results on
the multiplicity of solutions, and on their Morse indices. In particular, we
easily recover the classical results of D.D. Joseph and T.S. Lundgren [6] on
the Gelfand problem. Surprisingly, the situation turns out to be different for
the generalized Gelfand problem, where infinitely many turns are possible for
any space dimension $n \geq 3$. We also derive detailed results for the
equation modeling electrostatic micro-electromechanical systems (MEMS), in
particular we easily recover the main result of Z. Guo and J. Wei [4], and we
show that the Morse index of the solutions increases by one at each turn. We
also consider the self-similar Henon's equation.
