Existence and multiplicity of positive solutions for discrete anisotropic equations

In this paper we consider the Dirichlet problem for a discrete anisotropic equation with some function a , a nonlinear term f, and a numerical parameter l :D (a (k) |D u(k-1)|p(k-1)-2D u(k-1)) + l f(k,u(k))=0, k\in [1,T] . We derive the intervals of a numerical parameter l for which the considered BVP has at least 1, exactly 1, or at least 2 positive solutions. Some useful discrete inequalities are also derived.

Anahtar Kelimeler:

Discrete boundary value problem, variational methods, Ekeland's variational principle, mountain pass theorem, Karush--Kuhn--Tucker theorem, positive solution, anisotropic problem

Existence and multiplicity of positive solutions for discrete anisotropic equations

Keywords:

Discrete boundary value problem, variational methods, Ekeland's variational principle, mountain pass theorem, Karush--Kuhn--Tucker theorem, positive solution, anisotropic problem,

PDF

___

Agarwal RP, Perera K, O’Regan D. Multiple positive solutions of singular discrete p-Laplacian problems via variational methods. Adv. Difference Equ. 2 2005; 93–99.
Bereanu C, Jebelean P, S ¸erban C. Periodic and Neumann problems for discrete p( ·)−Laplacian. J. Math. Anal. Appl. 399, No. 1, 2013; 75–87.
Bian LH, Sun HR, Zhang QG. Solutions for discrete p − Laplacian periodic boundary value problems via critical point theory. J. Difference Equ. Appl. 18, No. 3 2012; 345–355.
Borwein JM, Lewis AS. Convex analysis and nonlinear optimization. Theory and examples. 2nd ed., CMS Books in Mathematics/Ouvrages de Math´ ematiques de la SMC 3. New York, NY: Springer. xii, 2006.
Cai X, Yu J. Existence theorems of periodic solutions for second-order nonlinear difference equations. Adv. Difference Equ. 2008, Article ID 247071.
Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image processing. SIAM J. Appl. Math. 2006; 66: 1383–1406.
Dajun G. Nonlinear Functional Analysis. Shandong Science and Technology Press, 1985.
De Figueiredo DG. Lectures on the Ekeland Variational Principle with Applications and Detours. Bombay: Tata Institute of Fundamental Research, 1989.
Galewski M, Gl¸ ab S, Wieteska R. Positive solutions for anisotropic discrete boundary-value problems. Electron. J. Differ. Equ. 2013, Paper No.32, 9 p., electronic only (2013).
Galewski M, Wieteska R. On the system of anisotropic discrete BVPs. to appear J. Difference Equ. Appl., DOI:1080/10236198.2012.709508.
Harjulehto P, H¨ ast¨ o P, Le UV, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Anal. 2010; 72: 4551–4574.
Kone B, Ouaro S. Weak solutions for anisotropic discrete boundary value problems. J. Difference Equ. Appl. 2011; 17: 1537–1547.
Liu JQ, Su JB. Remarks on multiple nontrivial solutions for quasi-linear resonant problemes. J. Math. Anal. Appl. 2001; 258: 209–222.
Mawhin J. Probl` emes de Dirichlet variationnels non lin´ e aires. Les Presses de l’Universit´ e de Montr´ eal, 1987.
Mihˇ ailescu M, Rˇ adulescu V, Tersian S. Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl. 2009;15: 557–567.
R˚ uˇ ziˇ cka M. Electrorheological fluids: Modelling and Mathematical Theory. in: Lecture Notes in Mathematics, vol. 1748, Berlin: Springer-Verlag, 2000.
Stehl´ık P. On variational methods for periodic discrete problems. J. Difference Equ. Appl. 2008; 14 (3): 259–273. Tian Y, Du Z, Ge W. Existence results for discrete Sturm-Liouville problem via variational methods. J. Difference Equ. Appl. 2007; 13 (6): 467–478.
Yang Y, Zhang J. Existence of solution for some discrete value problems with a parameter. Appl. Math. Comput. 2009; 211: 293–302.
Zhang G. Existence of non-zero solutions for a nonlinear system with a parameter. Nonlinear Anal. 2007; 66 (6): 1400–1416.
Zhang G, Cheng SS. Existence of solutions for a nonlinear system with a parameter. J. Math. Anal. Appl. 2006; 314 (1): 311–319.
Zhang X, Tang X. Existence of solutions for a nonlinear discrete system involving the p −Laplacian. Appl. Math. 2012, Praha 57, No. 1: 11–30.
Zhikov VV. Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR 1987; Izv. 29: 33–