Research On Invariant Sets Method And The Eigenvalues Of Degenerate Elliptic Operator

In this doctor dissertation, we mainly considered the relations between semi-dynamical systems and the equilibriums, which include how to find equilibriums by the semi-dynamical systems and how to seek positive invariant sets of the semi-flow.First, the discussion was carried out for the semi-groups with Lyapunov func-tionals on the attractors, because of its nature connection with the invariant sets of descending flow method in many equations. But it need to pay attention to the time-opposite ill-posedness of the semi-group.Then the following two contributions were made around the initial questions.Schauder condition, which is original defined for the convex sets of Hilbert spaces to describing the property of the boundary under the derivative of the func-tionals. was generalized to fit essentially local convex sets of Banach spaces,And a proper pseudo-gradient flow was constructed, under which the invariance was kep-t. Then the progress was applied to get the nontrivial solutions of the p-Laplace equations with zero and infinite has different growth rate nonlinear term.Benefit by the reinforced definition of the pseudo-gradient vector field, a global pseudo-gradient flow was constructed and proved for general C1 functional. Global means the maximum existence interval of the flow will be infinite, and it can be well defined for any initial points including the critical points set. It prevents the usual cut off steps, and more suitable to learn its dynamics.For further study, eigenvalues of weighted Laplace-Dirichlet operator was con-sidered. Especially when the imbedding of the weighted Sobolev spaces into the L2 is no longer compact, several modifications was made in the discussion. And that, the significance of this problem can be interpret as:this operator is a basic tool of describing the degenerate, nonhomogeneity or unbounded problems of evolution equations; more important is that, when dealing with some nonlinear problems such as p-Laplace equations, some part of the nonlinear operator many be considered as a weight to simplify the discussions.