Existence of Positive Solutions for Singular One-Dimensional $P$-Laplace BVP of the Second-Order Difference Systems

DOI：10.3770/j.issn:2095-2651.2013.02.006

 作者 单位 胡卫敏 伊犁师范学院数学与统计学院, 新疆 伊宁835000 巴哈尔古丽 伊犁师范学院数学与统计学院, 新疆 伊宁835000 蒋达清 东北师范大学数学与统计学院, 吉林 长春 130024

应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性，其中$\phi(s) = |s|^{p-2}s, ~p>1$，非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性.

In this paper we establish the existence of single and multiple positive solutions to the following singular discrete boundary value problem $$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~i\in \{1,2,\ldots,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.\tag 1.1$$ where $\phi(s)=|s|^{p-2}s$, $p>1$ and the nonlinear terms $f_{k}(i,x,y)~(k=1,2)$ may be singular at $(x,y)=(0,0)$.