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Keywords:
discrete $p$-Laplacian eigenvalue problem; positive solution; continuum; Picone-type identity; lower and upper solutions method
discrete $p$-Laplacian eigenvalue problem; positive solution; continuum; Picone-type identity; lower and upper solutions method
Summary:
We discuss the discrete $p$-Laplacian eigenvalue problem, \[ \begin {cases} \Delta (\phi _p(\Delta u(k-1)))+\lambda a(k)g(u(k))=0,\quad k\in \{1,2, \ldots , T\},\\ u(0)=u(T+1)=0, \end {cases} \] where $T>1$ is a given positive integer and $\phi _p(x):=|x|^{p-2}x$, $p > 1$. First, the existence of an unbounded continuum $\mathcal {C}$ of positive solutions emanating from $(\lambda , u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $\mathcal {C}$ is a monotone continuous curve globally defined for all $\lambda >0$.
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