Let G = V , E be a connected finite graph and Δ p be the p -Laplacian on G with p > 1 . We consider a perturbed p -th Yamabe equation − Δ p u − λ u p − 2 u = h u α − 2 u + ε f , where h , f : V ⟶ ℝ are functions with h , f > 0 ; 1 < p < α ; λ and ε are two positive constants. Using the variational method, we prove that there exists some positive constant ϵ 1 such that for all ϵ ∈ 0 , ϵ 1 , the above equation has two distinct solutions.

As the space-time model of the theory of relativity, four-dimensional Minkowski space is the basis of the theoretical framework for the development of the theory of relativity. In this paper, we introduce Darboux vector fields in four-dimensional Minkowski space. Using these vector fields, we define some new planes and curves. We find that the new planes are the instantaneous rotation planes of rigid body moving in four-dimensional space-time. In addition, according to some characteristics of Darboux vectors in geometry, we define some new space curves in four-dimensional space-time and describe them with curvature functions. Finally, we give some examples.