The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.
Antonella NASTASI
,
Stepan TERSIAN
,
Calogero VETRO
. HOMOCLINIC SOLUTIONS OF NONLINEAR LAPLACIAN DIFFERENCE EQUATIONS WITHOUT AMBROSETTI-RABINOWITZ CONDITION[J]. Acta mathematica scientia, Series B, 2021
, 41(3)
: 712
-718
.
DOI: 10.1007/s10473-021-0305-z
[1] Agarwal R P. Difference Equations and Inequalities:Methods and Applications. Second Edition. Revised and Expanded. New York:M Dekker Inc, Basel, 2000
[2] Cabada A, Iannizzotto A, Tersian S. Multiple solutions for discrete boundary value problems. J Math Anal Appl, 2009, 356:418-428
[3] Cabada A, Li C, Tersian S. On homoclinic solutions of a semilinear p-Laplacian difference equation with periodic coefficients. Adv Difference Equ, 2010, 2010:195376
[4] Diening L, Harjulehto P, Hästö P, Rǔzǐcka M. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math. Vol 2017. Heidelberg:Springer-Verlag, 2011
[5] Iannizzotto A, Tersian S. Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory. J Math Anal Appl, 2013, 403:173-182
[6] Jiang L, Zhou Z. Three solutions to Dirichlet boundary value problems for p-Laplacian difference equations. Adv Difference Equ, 2008, 2008:345916
[7] Kelly W G, Peterson A C. Difference Equations:An Introduction with Applications. New York, Basel:Academic Press, 1991
[8] Motreanu D, Motreanu V V, Papageorgiou N S. Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. New York:Springer, 2014
[9] Motreanu D, Vetro C, Vetro F. A parametric Dirichlet problem for systems of quasilinear elliptic equations with gradient dependence. Numer Func Anal Opt, 2016, 37:1551-1561
[10] Motreanu D, Vetro C, Vetro F. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete Contin Dyn Syst Ser S, 2018, 11:309-321
[11] Mugnai D, Papageorgiou N S. Wang's multiplicity result for superlinear (p, q)-equations without the Ambrosetti-Rabinowitz condition. Trans Amer Math Soc, 2014, 366:4919-4937
[12] Nastasi A, Vetro C. A note on homoclinic solutions of (p, q)-Laplacian difference equations. J Difference Equ Appl, 2019, 25:331-341
[13] Nastasi A, Vetro C, Vetro F. Positive solutions of discrete boundary value problems with the (p, q)-Laplacian operator. Electron J Differential Equations, 2017, 2017:225
[14] Papageorgiou N S, Vetro C, Vetro F. Multiple solutions with sign information for a (p, 2)-equation with combined nonlinearities, Nonlinear Anal, 2020, 192:111716
[15] Peral I. Multiplicity of solutions for the p-laplacian. Lecture Notes at the Second School on Nonlinear Functional Analysis and Applications to Differential Equations at ICTP of Trieste. 1997
[16] Pucci P, Serrin J. A mountain pass theorem. J Differential Equations, 1985, 60:142-149
[17] Saavedra L, Tersian S. Existence of solutions for nonlinear p-Laplacian difference equations. Topol Methods Nonlinear Anal, 2017, 50:151-167
