"Large" conformal metrics of prescribed Q-curvature in the negative case

Given a compact and connected four dimensional smooth Riemannian manifold (M, g(0)) with k(P) := integral(M) Q(g0)dV(g0) < 0 and a smooth non-constant function f(0) with max(p is an element of M) f(0)(p) = 0, all of whose maximum points are non-degenerate, we assume that the Paneitz operator is nonnegative and with kernel consisting of constants. Then, we are able to prove that for sufficiently small lambda > 0 there are at least two distinct conformal metrics g lambda = e(2u lambda)g(0) and g(lambda) = e(2u)lambda g(0) of Q-curvature Q(g lambda) = Q(g lambda) = f(0) + lambda. Moreover, by means of the "monotonicity trick" in a way similar to [9], we obtain crucial estimates for the "large" solutions u. which enable us to study their "bubbling behavior" as lambda down arrow 0.