Nonlinear boundary value problems : existence, estimates, symmetry

A differential equation may be seen as a condition relating the height, the slope, the curvature... of a curve or a (hyper-)surface. Most of the time, boundary conditions are imposed. Such problems appear in many areas of applied sciences, as well as in many mathematical problems, as for example, the Sobolev inequalities. In this thesis, we present some existence results for a class of ordinnary quasilinear differential equations, some bifurcation results around the solution 1 for the Lane-Emden partial differential equations with the Neumann boundary conditions, and prove the existence of an optimal function for the aforementioned inequalities with null mean value conditions, in the delicate BV case. We also focus on the symmetries or the breaking of symmetries of the so-called least energy nodal solutions of the Lane-Emden problem when its parameter is close to 2, in the delicate case of a non-radial domain. At last, we generalize this result to other problems dealing with non-homogeneous nonlinearities.(MATH 3) -- UCL, 200