Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35].

Given a compact surface (Σ,g), we prove the existence of a solution for the mean field equation on Σ. The problem consists of solving a second-order nonlinear elliptic equation with variational structure and exponential nonlinearity. Since the corresponding Euler functional is, in general, unbounded from above and from below, we employ topological methods and min-max schemes. Our result generalizes previous results by Lin [11], by Chen and Lin [6] and by Ding et al. [8]. The main point here is that, by taking values of the parameter greater than 8π, we make no assumption on the topology of the surface.

Following standard notations, · p in the above expression stands for the L p -norm (with respect to the Riemannian measure dv g ). As is well known and easy to see, for all u ∈ C ∞ (M), ( g u) 2 ≤ n|∇ 2 u| 2 .Conversely, by the Bochner-Lichnerowicz-Weitzenböck formula we get that DJADLI, HEBEY, AND LEDOUX

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere (S n , h). We derive from this analysis some a priori estimates in dimension 5 and 6. On S 5 these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On S 6 we prove the existence of at least one solution when an index formula associated to this conformal invariant is different from zero. Estimates for isolated simple blow up pointsIn this section we study the properties of isolated simple blow up points for equation (3). We first prove some Harnack type inequalities. In the following, given r > 0, B r will denote the open ball of radius r centred at 0 in R n , and B r its closure.