Abstract
In this paper, we investigate the solutions of boundary value problems for second-order p-Laplacian difference equations. By using the critical point theory, the existence and multiple results are obtained.
Highlights
In this paper, we consider the following second-order p-Laplacian difference equation: qnφp( xn–1) + fn(xn) = 0, n ∈ Z(1, k), (1.1)with boundary value conditions αx0 – β x0 = 0 = γ xk+1 + σ xk, (1.2)where is the forward difference operator xn = xn+1 – xn, φp(s) is the p-Laplacian operator φp(s) = |s|p–2s (1 < p < ∞), qn is real-valued for each n ∈ Z, k is a given positive integer, α, β, γ and σ are constants, f ∈ C(R2, R)
In 2007, Chen and Fang [6] obtained a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equation φp( xn–1) + fn(xn+1, xn, xn–1) = 0, n ∈ Z, (1.5)
By using critical point theory, Chen and Tang [9] established some existence criteria to guarantee that the second-order discrete p-Laplacian system φp( xn–1) – an|xn|p–2xn + ∇Wn(xn) = 0, n ∈ Z, (1.7)
Summary
1 Introduction In this paper, we consider the following second-order p-Laplacian difference equation: qnφp( xn–1) + fn(xn) = 0, n ∈ Z(1, k), (1.1) In 2007, Chen and Fang [6] obtained a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equation φp( xn–1) + fn(xn+1, xn, xn–1) = 0, n ∈ Z, (1.5) Shi et al [7] investigated the existence and multiplicity results for the Dirichlet boundary value problem (BVP) of (1.4) by using the mountain pass lemma in combination with variational techniques.
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