Global continuum of positive solutions for discrete \(p\)-Laplacian eigenvalue problems. (English) Zbl 1374.39005

The authors considered the following discrete \(p\)-Laplacian eigenvalue problem: \[ \Delta\left(\varphi_{p}\left(\Delta u(k-1)\right)\right)+\lambda a(k) g(u(k))=0,\quad k \in [1,T]_{\mathbb{Z}}\tag{EP\(_1\)} \] subject to the boundary conditions \[ u(0)=u(T+1)=0,\tag{EP\(_2\)} \] where \(T \in \mathbb{N}\), \(T>1\), \(\varphi_{p}(x):=|x|^p\), with \(p>1\), \(\Delta u(k):=u(k+1)-u(k)\), \(\lambda \geq 0\) and \([1,T]_{\mathbb{Z}}\) denotes the set \(\{1,2,\dots, T\}\). \(E\) denotes the set of functions \(u:[0,T+1]_{\mathbb{Z}} \rightarrow \mathbb{R}^{T+2}\) equipped with the norm \(\|u\|=\max\{|u(t)|: t \in [0,T+1]_{\mathbb{Z}}\}\). Given \(u,v\) in \(E\), \(u \leq v\) means that \(u(k) \leq v(k),\) \(\forall k \in [0,T+1]_{\mathbb{Z}}\) and \(u \prec v\) means that \(u \leq v\) and \(u(k)<v(k)\) for all \(k \in [1,T]_{\mathbb{Z}}\). The set \(K\) is the set of elements \(u \in E\) such that \(u \geq 0\) and \(\Delta^2u(k-1) \leq 0\) for all \(k \in [1,T]_{\mathbb{Z}}\). \(S\) is the set of positive solutions of (EP) in \([0,\infty) \times K\). The key results proved in the paper are the following two theorems.
Theorem 1: If \(a:[1,T]_{\mathbb{Z}} \rightarrow (0,\infty):=\mathbb{R}^{+}\), \(g:\mathbb{R}_{0}^{+} \rightarrow \mathbb{R}^{+}\) is continuous and \[ \lim_{u \rightarrow \infty}\frac{g(u)}{\varphi_{p}(u)}=0, \] then there exists an unbounded continuum \(\mathcal{C}\) of positive solutions for (EP) emanating from \((0,0)\) in \(\mathbb{R}_{0}^{+} \times K\) such that:
(i) \(\forall \lambda >0\), there exists a positive solution \(u(\lambda)\) of (EP) such that \((\lambda, u(\lambda)) \in \mathcal{C}\).
(ii) for \((\lambda, u(\lambda)) \in S\), \(\lambda \rightarrow \infty\) if and only if \(\|u(\lambda)\| \rightarrow \infty\).
Theorem 2: Under the same assumptions as in Theorem 1, if \(g(u)/\varphi_{p}(u)\) is strictly decreasing on \((0,\infty)\), then \(\mathcal{S}=\mathcal{C}\) and \(\mathcal{C}\) is the solution curve of positive solutions for (EP) such that:
(i) \(\forall \lambda >0\), there exists a positive solution \(u(\lambda)\) of (EP) such that \((\lambda, u(\lambda)) \in \mathcal{C}\).
(ii) \(\lambda \rightarrow \infty\) if and only if \(\|u(\lambda)\| \rightarrow \infty\).
(iii) \(\forall 0<\lambda_{a}<\lambda_{b}\), \(u(\lambda_{a}) \prec u(\lambda_{b})\).
As remarked by the authors, the proofs use results from [D. Bai, Adv. Difference Equ. 2013, Paper No. 264, 10 p. (2013; Zbl 1375.39009); D. Bai and X. Xu, Adv. Difference Equ. 2013, Paper No. 267, 13 p. (2013; Zbl 1375.39010); A. Cabada [Comput. Math. Appl. 42, No. 3–5, 593–601 (2001; Zbl 1001.39006)] and a simplified version of the generalized Picone identity due to [P. Řehák, Czech. Math. J. 51, No. 2, 303–321 (2001; Zbl 0982.39004)].


39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34L05 General spectral theory of ordinary differential operators
39A22 Growth, boundedness, comparison of solutions to difference equations
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