Radial Solutions of Semilinear Elliptic Equations with Prescribed Numbers of Zeros

Abstract

We study solutions u = u( t) of an initial value problem for u′′ + (( n − 1)/ t) u′ + ƒ( u) = 0. Under certain conditions on the nonlinearity ƒ (for instance ƒ( u) = −| u| q + | u| q̂−1 u, 1 < q < q̂ < ( n + 2)/( n − 2), n > 2), we get the existence of some initial values, such that the related solutions converge to zero after having a finite number of zeros. One principal tool to prove this result is derived from the consideration of the relation between critical points and succeeding zeros of solutions of this initial value problem for ƒ( u) = | u| q−1 u + o(| u| q) as | u| → 0.