Ground State Solutions of Discrete Asymptotically Linear Schrödinger Equations with Bounded and Non-periodic Potentials

• Published in: Journal of dynamics and differential equations Vol. 32; no. 2; pp. 527 - 555
• Main Authors: Lin, Genghong, Zhou, Zhan, Yu, Jianshe
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Asymptotic properties; Convergence; Ground state; Infinity; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear equations; Ordinary Differential Equations; Partial Differential Equations; Periodic variations; Schrodinger equation; Gap solitons; Saturable nonlinearity; Linking theorem; Variational methods; 39A12; 35Q55; Discrete nonlinear Schrödinger equations; Ground state solutions
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-019-09743-4

We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equations with a sign-changing potential V that converges at infinity and a nonlinear term being asymptotically linear at infinity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is that, due to the convergency of V at infinity, the classical methods such as periodic translation technique and compact inclusion method cannot be employed directly to deal with the lack of compactness of the Cerami sequence. New techniques are developed in this work to overcome these two major difficulties. This enables us to establish the existence of a ground state solution and derive a necessary and sufficient condition for a special case. To the best of our knowledge, this is the first attempt in the literature on the existence of a ground state solution for the strongly indefinite problem under no periodicity condition on the bounded potential and the nonlinear term being asymptotically linear at infinity. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation.