| Volume 15, Number 2, 2025, Pages - DOI：10.11948/JAAC-2024-0245 |
| Positive solutions of discrete boundary value problem for a second-order nonlinear difference equation with singular \phi-Laplacian |
| Ting Wang,Man Xu,Yanyun Li |
| Keywords:positive solutions, discrete boundary value problem, singular $\phi$-Laplacian, Lower and upper solutions, Szulkin"s critical point theory. |
| Abstract: |
| We establish the nonexistence, existence and multiplicity of positive solutions of the following discrete boundary value problem for a second-order nonlinear difference equation with singular $\phi$-Laplacian
{\footnotesize
\begin{equation*}
\begin{cases}
\begin{split}
&-\nabla(k^{N-1}\phi(\triangle v_k))=\lambda Nk^{N-1}\left(\frac{f"(\varphi^{-1}(v_k))}{\sqrt{1-(\triangle v_k)^2}}-f(\varphi^{-1}(v_k))H(\varphi^{-1}(v_k),k)\right),k\in[2,n-1]_{\mathbb{Z}},\&|\triangle v_k|<1,\&\triangle v_1=0=v_n,\\end{split}
\end{cases}
\end{equation*}}
where $\phi(s)=\frac{s}{\sqrt{1-s^2}}$, $\phi: (-1,1)\rightarrow\mathbb{R}$ is an increasing homeomorphism with $\phi(0)=0$, $\lambda$ is a positive parameter, $\triangle$ is the forward difference operator defined by $\triangle v_k=v_{k+1}-v_k$, $\nabla$ is the backward difference operator defined by $\nabla v_k=v_k-v_{k-1}$, $f\in C^{\infty}(I)$ and $f>0$, $I$ is an open interval in $\mathbb{R}$, $\varphi(s)=\int_0^s\frac{dt}{f(t)}$, $\varphi^{-1}$ is the inverse function of $\varphi$, $H:I\times[2,n-1]_{\mathbb{Z}}\rightarrow\mathbb{R}$ is a continuous function, $[2,n-1]_{\mathbb{Z}}:={2,3,\ldots,n-1}$, and the integer $n\geq4$. By using the method of upper and lower solutions, topological degree theory and Szulkin"s critical point theory for convex, lower semicontinous perturbations of $C^1$-functionals, we determine the interval of parameter $\lambda$ in which the above problem has zero, one, two positive solutions according to sublinear at zero. |
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