Electron. J. Differential Equations,
Vol. 2019 (2019), No. 96, pp. 114.
Homoclinic solutions of discrete nonlinear Schrodinger equations
with partially sublinear nonlinearities
Genghong Lin, Jianshe Yu, Zhan Zhou
Abstract:
We consider a class of discrete nonlinear Schrodinger (DNLS) equations
in m dimensional lattices with partially sublinear nonlinearity f.
Combining variational methods and a priori estimate, we give a general
sufficient condition on f for type (A), that is, a sequence of nontrivial
homoclinic solutions accumulating to zero.
By using a compact embedding technique, we overcome the loss of compactness
due to the problem being set on the unbounded domain
.
Another obstacle caused by the local definition of f is solved by using
the cutoff methods to recover the global property of f.
To the best of our knowledge, this is the first time to obtain infinitely
many homoclinic solutions for the DNLS equations with partially sublinear
nonlinearity. Moreover, we prove that if f is not sublinear, the zero
solution is isolated from other homoclinic solutions.
Our results show that the sublinearity and oddness of f yield type (A).
Without the oddness assumption, we still can prove that this problem has
at least a nontrivial homoclinic solution if f is local sublinear, which
improves some existing results.
Submitted October 2, 2018. Published August 2, 2019.
Math Subject Classifications: 39A12, 35Q55.
Key Words: Discrete nonlinear Schrodinger equation; discrete breathers;
homoclinic solution; partially sublinear nonlinearities;
variational method.
Show me the PDF file (343 KB),
TEX file for this article.

Genghong Lin
Center for Applied Mathematics
Guangzhou University
Guangzhou 510006, China
email: lin_genghong@163.com


Jianshe Yu
Center for Applied Mathematics
Guangzhou University
Guangzhou 510006, China
email: jsyu@gzhu.edu.cn


Zhan Zhou
Center for Applied Mathematics
Guangzhou University
Guangzhou 510006, China
email: zzhou0321@hotmail.com

Return to the EJDE web page