Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditions. (English) Zbl 1491.35405

Lin, Genghong; Yu, Jianshe

SIAM J. Math. Anal. 54, No. 2, 1966-2005 (2022).

Summary: We deal with the existence and multiplicity of homoclinic solutions for a class of periodic discrete Schrödinger equations with local superquadratic conditions. The arising problem intensely involves major difficulties including the indefiniteness of the associated variational problem, the restriction of local superquadratic conditions, and the boundedness of Cerami sequences. New methods including weak*-compactness and an approximation scheme are developed in this work to conquer these difficulties. This allows us to obtain a ground state solution and infinitely many geometrically distinct solutions. To the best of our knowledge, this is the first time in the existing literature to obtain the existence and multiplicity results of such a discrete problem under local superquadratic conditions. Even for the uniform superquadratic case, our results also significantly improve the well-known ones as special cases. Moreover, our weaker conditions may be suitable to other types of variational problems.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
39A12	Discrete version of topics in analysis
39A70	Difference operators
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
35A15	Variational methods applied to PDEs
37C29	Homoclinic and heteroclinic orbits for dynamical systems

Keywords:

periodic discrete nonlinear Schrödinger equation; homoclinic solution; ground state solution; local superquadratic condition; existence and multiplicity; critical point theory

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