Unilateral global structure for discrete Dirichlet problem with Minkowski-curvature operator

Abstract

In this paper, we are concerned with the unilateral global bifurcation results for the Minkowski-curvature discrete problem { − ∇ [ ϕ ( Δu ( t ) ) ] = λa ( t ) u ( t ) + f ( t , u ( t ) , λ ) , t ∈ [ 1 , T ] Z , u ( 0 ) = u ( T + 1 ) = 0 , where λ ∈ R is a parameter, a : [ 1 , T ] Z → ( 0 , ∞ ) , f ∈ C ( [ 1 , T ] Z × R 2 , R ) , Δ is the forward difference operator with Δ u ( t ) = u ( t + 1 ) − u ( t ) and ∇ is the backward difference operator with ∇ u ( t ) = u ( t ) − u ( t − 1 ) , ϕ ( s ) = s 1 − s 2 is a ϕ-Laplacian operator. We shall show that there are two distinct unbounded continua C + and C − , consisting of the bifurcation branch C if f is not necessarily differentiable at the origin with respect to u . As the applications of the above result, we shall investigate the existence and asymptotic behaviours of one-sign solutions for the half-quasilinear discrete problem { − ∇ [ ϕ ( Δu ( t ) ) ] = α ( t ) u + ( t ) + β ( t ) u − ( t ) + μa ( t ) F ( u ( t ) ) , t ∈ [ 1 , T ] Z , u ( 0 ) = u ( T + 1 ) = 0 , where μ ≠ 0 is a parameter, a : [ 1 , T ] Z → ( 0 , + ∞ ) , α , β : [ 1 , T ] Z → R , u + = max { u , 0 } , u − = − min { u , 0 } , F ∈ C ( R , R ) satisfies sF ( s ) > 0 for s ≠ 0 . We will use the obtained unilateral global bifurcation theory and the approximation of connected components to prove main results.