Exact Morse index of radial solutions for semilinear elliptic equations with critical exponent on annuli

Let N = 3, R >rho > 0 and A rho := {x is an element of R-N;rho < | x| < R}. Let U (+/-) (n,rho,) n >= 1, be a radial solution with n nodal domains of {Delta U + | x|(alpha)|U|Up-1 = 0 in A rho, U = 0 on partial derivative A rho We show that if p = N+2+2 alpha/ N-2, alpha > -2 and N >= 3, then U (+/-) (n,rho) is nondegenerate for small rho > 0 and the Morse index m(U (+/-) (n,rho)) satisfies m(U (+/-) (n,rho)) = n (N + 2l - 1)( N + l - 1)!/ ( N - 1)! l! for small rho > 0, where l = [ alpha/2] + 1. Using Jacobi elliptic functions, we show that if ( p, alpha) = (3, N - 4) and N >= 3, then the Morse index of a positive and negative solutions m(U (+/-) (1,rho)) is completely determined by the ratio rho/ R is an element of (0, 1). Upper and lower bounds for m(U (+/-) (n,rho)), n >= 1, are also obtained when ( p, alpha) = (3, N - 4) and N >= 3.