The existence of spherically symmetric solutions of the equation Au = /(u) is proved for a large class of functions f(z). Among others, functions satisfying an inequality zf(z) < 0 for \z\ > A, and in particular the function/(z) = -sinh z, belong to this class. 1980 Mathematics subject classification (Amer. Math. Soc.): 35 J 15, 35 J 60.