Abstract
In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ -Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.
Highlights
Let Z and R denote the sets of integers and real numbers, respectively
We extend the domain of the singular operator to the whole real numbers and consider an auxiliary problem associated with the original one
The conditions for the multiplicity of positive solutions of the discrete boundary problem are found, and an illustrative example is given
Summary
Let Z and R denote the sets of integers and real numbers, respectively. For a, b ∈ Z, define ZðaÞ = fa, a + 1,⋯g and Zð a, bÞ = fa, a + 1,⋯,bg when a ≤ b. We consider the following boundary value problem of prescribed mean curvature equations in Minkowski spaces:. Earlier in 2008, Bereanu and Mawhin in [2] obtained the existence of at least one or two solutions for the boundary value problems of second-order nonlinear differences with singular φ-Laplacian by using the Brouwer degree together with fixed point reformulations. For the existence and multiplicity of solutions of boundary value problems of difference equations, the classical methods are fixed point theory, the method of upper and lower solution techniques, Rabinowitz’s global bifurcation theorem, etc. Boundary value problems of difference equations involving φ-Laplacian have aroused extensive attention from scholars; for example, in 2019, Zhou and Ling in [28] considered the following Dirichlet problem of the second-order nonlinear difference equation:.
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