Abstract

In this paper we consider the discrete anisotropic difference equation with variable exponent using critical point theory. The study of nonlinear difference equations has now attracted special attention as they have important applications in various research areas such as numerical analysis, computer science, mechanical engineering, cellular neural networks and population growth, cybernetics, etc. In many studies, the authors consider Dirichlet, Neumann or Robin type boundary conditions. However, in this paper, we consider a homoclinic boundary condition, which means that the value of the solution is equal to a constant at infinity. Here we assume that the value of the solution vanishes at infinity. In this paper, we are also interested in the existence of at least one non-trivial homoclinc solution. To achieve this, we apply firstly the direct variational method and secondly the well-known Mountain pass technique, known as the Mountain pass theorem of Ambrosetti and Rabinowitz, to obtain the existence of at least one non-trivial homoclinic solution.

Keywords: Anisotropic difference equation; critical point theory; Mountain pass lemma; direct variational method.

MSC: 39A27, 39A05, 39A14, 39A60

DOI: 10.57016/MV-Kdn829ss

Pages:  1--14