Scalar curvature type equations on compact riemannian manifolds

Zindine Djadli Universite de Cergy-Pontoise Departement de Mathematiques Site de Saint-Martin 2 avenue Adolphe Chauvin F-95302 Cergy-Pontoise Cedex France

                djadli@paris.u-cergy.fr

Taking (M, g) a compact riemannian n-manifold, a, f and h three smooth functions on M and q ∈ (1, \frac{n+2}{n−2}), we study the following problem (P)

Δ_{g}u + au = fu^{\frac{n+2}{n−2}} + hu^{q}
u ∈ C^{∞}(M) , u > 0 on M

where Δ_{g} is the laplacian with the minus sign convention. In the first part, we want to find some conditions on a, f and h for P to have a solution; we distinguish two cases (Δ_{g}+a is coercive and Δ_{g}+a is not coercive) : we give, in each case, some existence results. The second part is devoted to uniqueness problem. Finally, in the third part, we give some existence results in the case where the exponent \frac{n+2}{n−2} is replaced by p > \frac{n+2}{n−2} (super-critical case).