Now, we recall an abstract linking theorem which is from [1, 12, 15].
In this context, two or more abstract links, which use the same physical links, may have correlated latencies [3], bandwidth usage [4], or geographical failures [5, 6].
For example, in overlay [1] or multilayer networks [2], the abstract links in the logical layer are mapped to different links in the physical layer.
For this matter, first we get two ground states in view of the Nehari procedure, establish a linking structure by the abstract linking theorem on the product space, and then construct the third critical value by the classical minimax principle.
Applying the Nehari procedure and abstract linking theorem on a product space, we give a linking structure for our variational problem, and then, combining with the classical minimax principle, we obtain three nontrivial critical values for the relevant energy functional.
In [9], based on an abstract linking theorem for smooth functionals, they also established a multiplicity result on the existence of weak solutions for a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator.
In this section, we list some preliminaries, including the abstract linking theorem on the product space and the concept of Nehari manifold, and then we give a linking structure, which is useful for constructing the critical value of the functional associated with system (1.1).
The ontology fingerprints were derived from 178,687 abstracts linked to at least one human gene.
The well-annotated abstracts linked to the GO terms are obtained from various sources such as SwissProt, GenBank, and FlyBase etc.
