Multiple positive solutions of second-order nonlinear difference equations with discrete singular ϕ-Laplacian

• Main Authors: Xiaoxiao Su, Ruyun Ma
• Format: Article
• Language: English
• Published: SpringerOpen 2020-12-01
• Series: Advances in Difference Equations
• Subjects: Discrete ϕ-Laplacian; Positive solutions; Multiplicity; Topological method
• Online Access: https://doi.org/10.1186/s13662-020-03135-5

Abstract We consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation { − ∇ [ ϕ ( △ u ( t ) ) ] = λ a ( t , u ( t ) ) + μ b ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( N ) = 0 , $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ where λ , μ ≥ 0 $\lambda ,\mu \geq 0$ , T = { 2 , … , N − 1 } $\mathbb{T}=\{2,\ldots ,N-1\}$ with N > 3 $N>3$ , ϕ ( s ) = s / 1 − s 2 $\phi (s)=s/\sqrt{1-s^{2}}$ . The function f : = λ a ( t , s ) + μ b ( t , s ) $f:=\lambda a(t,s)+\mu b(t,s)$ is either sublinear, or superlinear, or sub-superlinear near s = 0 $s=0$ . Applying the topological method, we prove the existence of either one or two, or three positive solutions.