| This thesis is devoted to the study of some quasi-linear PDEs involving the p-Laplacian. This type of problem represents a model case for the general quasi-linear elliptic equations. These problems arise from the Euler-Lagrange equations associated to various geometric problems and from topics like Non-Newtonian Fluids, Air Dynamics, Non-linear Biological Population, Reaction-Diffusion Problems etc. The difficulties in these problems come either from the lack of compactness of the approximate solutions or from the lack of symmetry in the corresponding energy functionals.; The thesis has basically two parts. The first part is in Chapter 3, where we consider a problem related to Yamabe's prescribed curvature conjecture in Riemannian Geometry. This problem is critical in the sense that it involves the critical exponent in the Sobolev embedding. In particular, we show the existence of sign changing solutions by using the duality methods introduced by N. Ghoussoub.; In the second part of the thesis, chapters 4-7, we consider the associated non-homogeneous problems which arise from either a linear second member or from non-homogeneous Dirichlet boundary conditions. Because of the lack of symmetry, the traditional equivariant variational principles do not apply here. To overcome those difficulties, we extend and use Bolle's method as well as Ekeland-Ghoussoub's virtual critical point theory to a Banach space setting. |
