Critical points of locally Lipschitz functionals inside and outside the ordered interval

Abstract

In this paper, under the condition that there exists an ordered interval composed of two internal ordered intervals which have the location similar to that of Amann's three-solution theorem, we add some simple conditions and then we obtain some results about the existence of multiple critical points inside and outside the ordered interval. The main results of this paper can be regarded as an extension of the classical Amann three-solution theorem and the mountain pass lemma on the ordered interval of Shujie Li and Zhiqiang Wang. To show our main results, we extend the method of invariant sets of descending flow that proposed by Jingxian Sun for smooth functionals to the locally Lipschitz functionals. Our main results can be applied to the study of differential inclusion problems with concave-convex nonlinearity. In this way, we partially extend some relevant results concerning the differential equation boundary value problems with a concave-convex nonlinearity that was first studied by A. Ambrosetti, H. Brezis and G. Cerami.

Keywords

• locally lipschitz functionals
• critical points
• differential inclusion problems