Abstract
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 q 2⊂ L p with 1⩽ q< n/2 and p= nq/( n−2 q). μ being the inf of the functional associated to the equation (E) with f some constant, we have always K 2 2 μ⩽1 and if K 2 2 μ<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.
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