New Linking Theorems

The linking theorem is one of the most celebrated results in modern theory ofthe calculus of variations. It give the su?cient conditions for the existence of a (PS)sequence for a continuously di?erentiable functional, which consequently gives us acritical point if we assume the (PS) compactness condition. In this article, we aregoing to discuss whether the link theorem still hold under new conditions.We find that, by adopting the concept of critical point for a locally Lipschitz con-tinuous functional, if we add the corresponding (PS ) condition, we are able to general-ize the linking theorem for a C1 functional to the case of a locally Lipschitz continuousfunctional; We can also prove the linking theorem for a locally Lipschitz continuousfunctional under a new compactness condition: the (?-PS ) condition, which is weakerthan the (PS ) condition.We also try to make a new approach to study the linking theorem, which is di?er-ent from that stated above. In order to do this, we introduce the concept of invariantset, and gives a linking theorem which guarantees the existence of a (PS ) sequencein a closed convex set for a C1 functional, with some assumptions on the form of thefunctional under consideration. In the last part of this article, still utilizing the conceptof invariant set, but with a new line of thinking, we are able to prove the existence ofa (PS ) sequence for a Locally Lipschitz continuous functional in a closed convex set,without specifying the functional form.