In this paper, we introduce some aspects of a critical point theory for multivalued functions Φ : E → RN∪{∞} defined on E a complete gauge space and with closed graph. The existence of a critical point is established in presence of linking. Finally, we present applications of this theory to semilinear elliptic problems on RN .

A gain graph (Γ, g, G) is a graph Γ = (V,E) together with a group G, the gain group, and a homomorphism g, the gain function, from the free group F (E) on the edge set E into G. We think of the edges of G as oriented in an arbitrary but fixed way, so that if e is an edge in one direction, then e−1 is the same edge in the opposite direction; thus g(e−1) = g(e)−1. A gain graph is balanced if for ...

a emph{signed graph} (or, in short, emph{sigraph}) $s=(s^u,sigma)$ consists of an underlying graph $s^u :=g=(v,e)$ and a function $sigma:e(s^u)longrightarrow {+,-}$, called the signature of $s$. a emph{marking} of $s$ is a function $mu:v(s)longrightarrow {+,-}$. the emph{canonical marking} of a signed graph $s$, denoted $mu_sigma$, is given as $$mu_sigma(v) := prod_{vwin e(s)}sigma(vw).$$the li...