Existence results to a nonlinear p(k)-Laplacian difference equation

Abstract

Abstract. In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno. IntroductionThe main goal of the present paper is to establish the existence of non-trivial solution for the following discrete anisotropic problem We want to remark that problem (1.1) is the discrete variant of the variable exponent anisotropic problem2) where Ω ⊂ R N , N ≥ 3 is a bounded domain with smooth boundary, f ∈ C(Ω×R, R) is given function that satisfy certain properties and p i (x), w i (x) ≥ 1 and q(x) ≥ 1 are continuous functions on Ω with 2 ≤ p i (x) for each x ∈ Ω and every i ∈ 2000 Mathematics Subject Classification. 39F20, 34B15.

Infinitely Many Solutions for Partial Discrete Kirchhoff Type Problems Involving p-Laplacian

“…In 2003, for the first time, Yu and Guo [7] investigated a class of difference equations. Since then, numerous researchers have studied difference equations and achieve a lot of findings, including the findings of periodic solutions [7][8][9][10], homoclinic solutions [11][12][13][14][15][16][17][18][19][20][21][22], and boundary value problems [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”