Abstract
Abstract Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory. We further separately determine open intervals of the parameter to attain at least two positive solutions and an unbounded sequence of positive solutions with the help of the maximum principle.
Highlights
Let Z and R signify all integers and real numbers, respectively
A partial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper
Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory
Summary
Let Z and R signify all integers and real numbers, respectively. For a, b ∈ Z, de ne Z(a, b) = {a, a+ , · · · , b} when a ≤ b and Z(a) = {a, a + , · · · }. ∆ (φc(∆ u(i − , j))) + ∆ (φc(∆ u(i, j − ))) + λf ((i, j), u(i, j)) = , (i, j) ∈ Z( , m) × Z( , n), with boundary conditions u(i, ) = u(i, n + ) = , i ∈ Z( , m + ), u( , j) = u(m + , j) = , j ∈ Z( , n + ), where λ is a positive real parameter, m and n are two given positive integers, ∆ and ∆ are two rst-order forward di erence operators, respectively given by ∆ u(i, j) = u(i+ , j)−u(i, j) and√∆ u(i, j) = u(i, j+ )−u(i, j), φc is a φ-Laplacian operator (mean curvature operator [1]) de ned by φc(s) = s/ + s , and for each (i, j) ∈ Z( , m) × Z( , n), f ((i, j), ·) ∈ C(R, R). In 2003, for the rst time, Guo and Yu [13] considered periodic solutions and subharmonic solutions for a class of di erence equations by using critical point theory. Many researchers have studied di erence equations via critical point theory so that a lot of excellent results have been. Sijia Du, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P.R.China Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P.R.China *Corresponding Author: Zhan Zhou, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P.R.China Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, P.R.China
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