On the monotonicity of the time-map

Abstract

The equation Au(y) + @(u(y)) = 0 on the unit ball, with homogeneous linear boundary conditions has been much studied (cf. [ 1,2 and the references therein]). The change of variable x= ,l;i!: transforms this equation to Mx) +.A+)) = 0, 1x1 <R, (1.1) where R = fi. For radial solutions of this equation, lying in, say, the kth nodal class, R depends only on p = u(O), R = T(p). This function, which describes how the “size of the balls” on which radial solutions exist, vary with the quantity u(O), turns out to be an important function, determining much of the behavior of the solutions. Thus, if we consider (1.1) together with the boundary conditions cu(x) - /I du(x)/dn = 0, 1x1 = R