## 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)].

### MSC:

 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

### Citations:

Zbl 1001.39006; Zbl 0982.39004; Zbl 1375.39009; Zbl 1375.39010
Full Text:

### References:

  Agarwal, R. P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular discrete $$p$$-Laplacian problems via variational methods, Adv. Difference Equ., 2005, 93-99, (2005) · Zbl 1098.39001  D. Bai: A global result for discrete $$φ$$-Laplacian eigenvalue problems. Adv. Difference Equ. 2013 (2013), Article ID 264, 10 pages. · Zbl 1375.39009  D. Bai, X. Xu: Existence and multiplicity of difference $$φ$$-Laplacian boundary value problems. Adv. Difference Equ. 2013 (2013), Article ID 267, 13 pages. · Zbl 1375.39010  Bian, L.-H.; Sun, H.-R.; Zhang, Q.-G., Solutions for discrete $$p$$-Laplacian periodic boundary value problems via critical point theory, J. Difference Equ. Appl., 18, 345-355, (2012) · Zbl 1247.39004  Cabada, A., Extremal solutions for the difference $$φ$$-Laplacian problem with nonlinear functional boundary conditions, Comput. Math. Appl., 42, 593-601, (2001) · Zbl 1001.39006  Jaroš, J.; Kusano, T., A Picone type identity for second-order half-linear differential equations, Acta Math. Univ. Comen., New Ser., 68, 137-151, (1999) · Zbl 0926.34023  Ji, D.; Ge, W., Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with $$p$$-Laplacian, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 68, 2638-2646, (2008) · Zbl 1145.34309  C.-G. Kim, J. Shi: Global continuum and multiple positive solutions to a $$p$$-Laplacian boundary-value problem. Electron. J. Differ. Equ. (electronic only) 2012 (2012), 12 pages. · Zbl 1260.34045  Kusano, T.; Jaroš, J.; Yoshida, N., A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 40, 381-395, (2000) · Zbl 0954.35018  Lee, Y.-H.; Sim, I., Existence results of sign-changing solutions for singular one-dimensional $$p$$-Laplacian problems, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 68, 1195-1209, (2008) · Zbl 1138.34010  Li, Y.; Lu, L., Existence of positive solutions of $$p$$-Laplacian difference equations, Appl. Math. Lett., 19, 1019-1023, (2006) · Zbl 1125.39007  Liu, Y., Existence results for positive solutions of non-homogeneous BVPs for second order difference equations with one-dimensional $$p$$-Laplacian, J. Korean Math. Soc., 47, 135-163, (2010) · Zbl 1191.39008  Řehák, P., Oscillatory properties of second order half-linear difference equations, Czech. Math. J., 51, 303-321, (2001) · Zbl 0982.39004  Xia, J.; Liu, Y., Positive solutions of BVPs for infinite difference equations with one-dimensional $$p$$-Laplacian, Miskolc Math. Notes, 13, 149-176, (2012)  Yang, Y.; Meng, F., Eigenvalue problem for finite difference equations with $$p$$-Laplacian, J. Appl. Math. Comput., 40, 319-340, (2012) · Zbl 1295.39006  E. Zeidler: Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Springer, New York, 1986. · Zbl 0583.47050  Zhang, X.; Tang, X., Existence of solutions for a nonlinear discrete system involving the $$p$$-Laplacian, Appl. Math., Praha, 57, 11-30, (2012) · Zbl 1249.39009
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