Abstract
We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms.
Highlights
Discrete boundary value problem is one of the most important mathematical equations and has rich applications in the area such as astrophysics, gas dynamics, fluid mechanics, computer science, image processing, chemically reacting systems, and others
The discrete p-Laplacian, which appears in various discrete problems, has received great attention from many researchers
We introduce a comparison principle and the sub-supersolution method for the discrete p-Laplacian with potential terms which are proved in [7, 8]
Summary
Discrete boundary value problem is one of the most important mathematical equations and has rich applications in the area such as astrophysics, gas dynamics, fluid mechanics, computer science, image processing, chemically reacting systems, and others. In [1], Agarwal, Perera and O’Regan proved the existence of multiple positive solutions to the following boundary value problem involving the discrete p-Laplacian:. −D(φp(Du(k − 1))) = λφp(u(k)), k ∈ [1, T ] u(0) = u(T + 1) = 0 (1.1) has a solution Their second result is that if f satisfies (1.2) and (1.4). We consider a discrete boundary value problem including potential terms on a graph. If (H1), (H2), and (H3)′ hold, (1.6) has a positive solution u1 satisfying u1 < w0. In addition, (H4) holds, (1.6) has the second positive solutions u2 satisfying u1 < u2.
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