Variational Approaches to a Discrete Elliptic Problem with a Weight

Abstract

In this article, we are concerned with the existence of at least one, two and three distinct solutions for discrete boundary value problems driven by the Laplacian. The proof of the main result depends on variational methods. By using a consequence of the local minimum theorem due Bonanno we investigate the existence of at least one solution and two solutions for the problem with the weight. Furthermore, by using two critical point theorems, one due Averna and Bonanno, and another due Bonanno we explore the existence of two and three solutions for the problem. We also provide two examples in order to illustrate the main results.

MATHEMATICS SUBJECT CLASSIFICATION:

• 39A10
• 35B38
• 30G25
• 47J30

Keywords:

• Critical point
• discrete elliptic problem
• nonlinear difference equation
• variational methods
• weight function