POSITIVE SOLUTIONS TO SEMILINEAR POLYHARMONIC DIRICHLET PROBLEMS INVOLVING CRITICAL SOBOLEV EXPONENTS

We are concerned with the semilinear polyharmonic model problem (-Delta)(K) upsilon = lambda upsilon + upsilon\upsilon\(s-1) in B, D(alpha)upsilon\partial derivative B = 0 for \alpha\ less than or equal to K - 1. Here K is an element of N, B is the unit ball in R(n), n > 2K, s = n+2K/n-2K is the critical Sobolev exponent. Let lambda(1) denote the first Dirichlet eigenvalue of (-Delta)(K) in B. The existence of a positive radial solution upsilon is shown for - lambda is an element of (0, lambda(1)), if n greater than or equal to 4K, - lambda is an element of (($) over bar lambda, lambda(1)) for some ($) over bar lambda = ($) over bar lambda(n, K) is an element of (0, lambda(1)), if 2K + 1 less than or equal to n less than or equal to 4K - 1. The crucial point of the present note is to show the positivity of a solution upsilon with help of the positivity of Green's function for (-Delta)(K) in balls.