Fourth-order elliptic equations with critical exponents on compact Riemann varieties

On a compact Riemannian manifold (M-n, g) with n > 4, to solve some elliptic equations of the fourth-order with critical Sobolev exponent we have first to precise the best constant K-q in the Sobolev inequality H-2(q) subset of L-p with 1 less than or equal to q < n/2 and p = nq/(n - 2q). mu being the inf of the functional associated to the equation (E) with f some constant, we have always K-2(2) mu < 1 and if K-2(2) mu < 1 the equation (E) has a non-zero solution. Some geometrical applications of this theorem are given. In some cases the solution is strictly positive. (C) 2001 Editions scientifiques et medicales Elsevier SAS.