EXISTENCE AND MULTIPLICITY OF SIGN-CHANGING SOLUTIONS FOR THE DISCRETE PERIODIC PROBLEMS WITH MINKOWSKI-CURVATURE OPERATOR
Zhongzi Zhao, Ruyun Ma, Yan Zhu · Journal of Applied Analysis & Computation · 2021
We are concerned with the discrete periodic problems with Minkowski-curvature operator \begin{document} $ \left\{\begin{array}{l} - abla(\frac{\Delta u(t)}{\sqrt{1-(\Delta u(t))^2}}) = \lambda g(t, u(t)), \quad t\in \mathbb{T}, \\ u(0) = u(T), u(1) = u(T+1), \end{array} \right. \qquad\qquad\qquad (P) $ \end{document} where $ T>2 $ is an integer, $ \mathbb{T}: = \{1, 2, \cdots, T\} $, $ \mathbb{\hat{T}} = \{0, 1, \cdots, T, T+1\} $, $ \mathbb{Z}: = \{\cdots, -1, 0, 1, \cdots\} $, $ g:\mathbb{Z}\times \mathbb{R}\rightarrow \mathbb{R} $ is continuous, $ g_0: = \lim\limits_{s\rightarrow0}\frac{g(t, s)}{s} = +\infty $ uniformly for $ t\in \mathbb{\hat T} $, $ g $ is $ T $-periodic respect to $ t $, $ \lambda\in (0, \infty) $ is a parameter. We show that $ (P) $ has multiple odd sign-changing solutions and multiple even sign-changing solutions when $ g(t, \cdot) $ is odd. The proof of our main result is based upon bifurcation techniques.