Positive solutions of the periodic problems for quasilinear difference equation with sign-changing weight

Abstract

Abstract We show the existence of positive solutions of the periodic problem of the quasilinear difference equation {−∇[ϕ(Δuk)]+qkuk=λgkf(uk),k∈T,u0=uT,u1=uT+1, $$\textstyle\begin{cases} -\nabla[\phi(\Delta u_{k})]+q_{k}u_{k}=\lambda g_{k}f(u_{k}),\quad k\in\mathbb {T},\\ u_{0}=u_{T}, \qquad u_{1}=u_{T + 1}, \end{cases} $$ where T={1,2,…,T} $\mathbb{T}=\{1, 2,\ldots,T\}$ with integer T≥2 $T\geq2$, ϕ(s)=s/1−s2 $\phi (s)=s/{\sqrt{1-s^{2}}}$, q=(q1,…,qT)∈RT $\boldsymbol{q}=(q_{1},\ldots,q_{T})\in\mathbb{R}^{T}$, qk≥0 $q_{k}\geq0$ for all k∈T $k\in\mathbb{T}$ and qk0>0 $q_{k_{0}}>0$ for some k0∈T $k_{0}\in \mathbb{T}$, g=(g1,…,gT)∈RT $\boldsymbol{g}=(g_{1},\ldots,g_{T})\in\mathbb{R}^{T}$ changes the sign on T $\mathbb{T}$, f is a continuous function, and λ∈R $\lambda\in\mathbb {R}$ is a parameter. The proofs of the main results are based upon bifurcation techniques.

Keywords

• Quasilinear difference equation
• Periodic problem
• Sign-changing weight
• Positive solution
• Bifurcation
• Spectrum