Abstract
Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.
Highlights
Denote Z and R the sets of integers and real numbers, respectively
We show that the suitable oscillating behavior of the nonlinear term near at the origin and at in nity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions
The results show that the suitable oscillating behavior of the nonlinear term f near at the origin and at in nity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions for problem (1.1)
Summary
Denote Z and R the sets of integers and real numbers, respectively. We consider the following Dirichlet problem of the second order nonlinear di erence equation. U = uT+ = , where T is a given positive integer, is the forward di erence operator de ned by uk = uk+ − uk, uk =. S +s [1], λ is a real positive parameter, and f (k, ·) ∈ C(R, R) for each k ∈ Z( , T). When qk ≡ , problem (1.1) may be regarded as the discrete analog of the following one-dimensional prescribed curvature problem. In [3], Bonanno, Livrea and Mawhin obtained an explicit interval Λ of positive parameters, such that, for every λ ∈ Λ, problem
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