| In Chapter 1, a sharp form for a pointwise estimate of blow-up solutions of a p-Laplacian type equation, which has been used among other things in the proof of the validity of optimal Sobolev inequalities on complete Riemannian manifolds, is obtained through the comparison principle and the careful choice of a test function.;Chapter 2 shows several ranges in which first order optimal Sobolev inequalities are not valid in Riemannian manifolds with positive scalar curvature somewhere, by showing that the associated Sobolev quotient becomes arbitrarily large for a family of minimizers of the Sobolev quotient localized at a point where the scalar curvature is positive. A comparison is made between those ranges where the optimal inequalities are known to be valid and those where we found they are not, showing that there remain a few ranges where it is not known if the optimal inequalities hold or not.;In Chapter 3, the best constants for Sobolev trace inequalities on Riemannian manifolds with boundary are established for any 1 < p < n. A version of the concentration-compactness principle for manifolds with boundary and the almost everywhere convergence of the gradients of solutions of a p-Laplace type equation are used in the proof.;In Chapter 4, the best constant for certain second order Sobolev inequalities on compact Riemannian manifolds with or without boundary, and its application to the resolution of fourth order nonlinear elliptic partial differential equations with critical exponent are studied. More precisely, if ( M, g) is a smooth compact Riemannian manifold, with or without boundary, of dimension n ≥ 3 and 1 < p < n2 , the norm u= Dgu pLpM +u pLpM 1/p is considered on each of the spaces H 2,p(M), H2,p0 (M) and H2, p(M) ∩ H1,p0 (M). The existence of an asymptotically sharp inequality associated to the critical Sobolev embedding of these spaces is shown. The non-validity of the associated optimal inequality for Riemannian manifolds with positive scalar curvature somewhere in the p = 2 case is also proven. As an application of the asymptotically sharp inequality, the influence of the geometry in the existence of solutions for some fourth order problems involving critical exponents on manifolds is investigated. In particular, new phenomena arise in Brezis-Nirenberg type problems on manifolds with positive scalar curvature somewhere, in contrast with the Euclidean case. The existence of solutions is proven through a suitable version of the concentration-compactness principle and issues of regularity of solutions are addressed in some cases. |
