  Using a variational approach, we obtain the self-focusing critical power for
a single and for any number of interacting Laguerre-Gauss beams propagating in
a Kerr nonlinear optical medium. As is known, the critical power for freely
propagating higher-order modes is always greater than that of the fundamental
Gaussian mode. Here, we generalize that result for an arbitrary incoherent
superposition of Laguerre-Gauss beams, adding interactions between them. This
leads to a vast and rich spectrum of self-focusing phenomena, which is absent
in the single-beam case. Specifically, we find that interactions between
different modes may increase or decrease the required critical power relative
to the sum of individual powers. In particular, high-orbital angular momentum
modes can be focused with less power in the presence of low-orbital angular
momentum beams than when propagating alone. The decrease in required critical
power can be made arbitrarily large by choosing the appropriate combinations of
modes. Besides, in the presence of interactions, an equilibrium configuration
of stationary spot-size for all modes in a superposition may not even exist, a
fundamental difference to the single-beam case in which a critical power for
self-focusing always exists.
