Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type

Abstract

We study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator ΔΔu(k−1)1−(Δu(k−1))2+λμ(k)(p+1)up(k)vq+1(k)=0,k∈[2,n−1]Z,ΔΔv(k−1)1−(Δv(k−1))2+λμ(k)(q+1)up+1(k)vq(k)=0,k∈[2,n−1]Z,Δu(1)=u(n)=0=Δv(1)=v(n),\left\{\begin{array}{ll}\Delta \left[\frac{\Delta u\left(k-1)}{\sqrt{1-{\left(\Delta u\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(p+1){u}^{p}\left(k){v}^{q+1}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta \left[\frac{\Delta v\left(k-1)}{\sqrt{1-{\left(\Delta v\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(q+1){u}^{p+1}\left(k){v}^{q}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta u\left(1)=u\left(n)=0=\Delta v\left(1)=v\left(n),& \\ \end{array}\right. where n∈Nn\in {\mathbb{N}} with n>4n\gt 4, max{p,q}>1\max \left\{p,q\right\}\gt 1, λ>0\lambda \gt 0, Δu(k−1)=u(k)−u(k−1)\Delta u\left(k-1)=u\left(k)-u\left(k-1), and μ(k)>0\mu \left(k)\gt 0 for all k∈[2,n−1]Zk\in {\left[2,n-1]}_{{\mathbb{Z}}}. The existence of zero at least one or two positive solutions for the system are obtained according to the different intervals of λ\lambda . Our main tools are based on topological methods, critical point theory, and lower and upper solutions.

Keywords

• discrete systems
• minkowski curvature operator
• positive solution
• brouwer degree
• lower and upper solutions
• critical point theory
• 39a13
• 47h11