Existence For Nonlinear Differential Equations

As everyone knows, it is of great importance to study existence and uniqueness for differential equations, There's many methods to study this, such as variational method, homeomorphism method, semigroup of operators etc. In this thesis, we discuss the existence and uniqueness for two kinds of nonlinear differential equations. First, with Galerkin approximation procedure and minimax principle, the existence and uniqueness for the weak solution of a nonlinear elliptic system without resonance is proved. At each finite dimensional step, we prove the existence of an approximate solution by applying a minimax theorem. Then we give an estimate for the approximate solutions. The fact that the operator with Dirichlet boundary value condition has a compact inverse gives us the existence and uniqueness for the weak solution. Secondly, we give a non variational version of minimax principle, and apply it in a kind of generic 2k th order ordinary differential equations with resonance and Duffing equations with three kinds of boundary value conditions.