Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator

Abstract

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

Positive Solutions for Dirichlet BVP of PDE Involving \({\varphi_{p}}\)-Laplacian

“…Lately, such equations have been widely used in many fields. The study of PDE is a challenging problem, gaining the attention of many researchers who have obtained some results [29][30][31][32][33][34][35][36][37][38].…”

“…In [28], the authors studied infinitely many solutions for the discrete BVP of the Kirchhoff type, and the problem only contains one variable. In [32], the authors considered the three solutions of the pde. In [41], the authors considered a class of fractional q-difference equations.…”