Abstract
This paper is concerned with positive solutions of the semilinear polyharmonic equation (-Delta)(m)u = a(x)u(alpha) on R-n, where m and n are positive integers with n > 2m, alpha is an element of (-1, 1). The coefficient a is assumed to satisfy
a(x) approximate to (1 + vertical bar x vertical bar)(-lambda) L(1 + vertical bar x vertical bar) for x is an element of R-n,
where lambda is an element of [2m, infinity) and L is an element of C-1([1, infinity)) is positive with tL'(t)/L(t) -> 0 as t -> infinity; if lambda = 2 m, one also assumes that integral(infinity)(1) t(-1)L(t)dt < infinity. We prove the existence of a positive solution u such that
u(x) approximate to (1 + vertical bar x vertical bar)(-<(lambda)over tilde>)(L) over tilde (1 + vertical bar x vertical bar) for x is an element of R-n,
with (lambda) over tilde := min(n - 2m, lambda-2m/1-alpha) and a function (L) over tilde, given explicitly in terms of L and satisfying the same condition at infinity. (Given positive functions f and g on R-n, f approximate to g means that c(-1) g <= f <= cg for some constant c > 1.)