BIFURCATIONS OF SOME ELLIPTIC PROBLEMS WITH A SINGULAR NONLINEARITY VIA MORSE INDEX

摘要

We study the boundary value problem Delta u = lambda vertical bar x vertical bar(alpha) f(u) in Omega, u = 1 on partial derivative Omega (1) where lambda > 0, alpha >= 0, Omega is a bounded smooth domain in R(N) (N >= 2) containing 0 and f is a C(1) function satisfying lim(s -> 0+) s(p) f(s) = 1. We show that for each alpha >= 0, there is a critical power p(c)(alpha) > 0, which is decreasing in alpha, such that the branch of positive solutions possesses infinitely many bifurcation points provided p > p(c)(alpha) or p > p(c)(0), and this relies on the shape of the domain Omega. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that p > p(c)(alpha) and the Morse index of any radial solution (regular or singular) in this branch is finite provided that 0 < p <= p(c)(alpha). This implies that the structure of the radial solution branch of (1) changes for 0 < p <= p(c)(alpha) and p > p(c)(alpha).