In this article, we deal with the nonlinear Schrödinger equation with nonlocal regional diffusion 0.1 (Formula presented.) where 0 < ? < 1, n ? 2, and (Formula presented.) is a continuous function. The operator (Formula presented.) is a variational version of the nonlocal regional Laplacian defined as (Formula presented.) where (Formula presented.) be a positive function. Considering that ?, V, and f(·, t) are periodic or asymptotically periodic at infinity, we prove the existence of ground state solution of (1) by using Nehari manifold and comparison method. Furthermore, in the periodic case, by combining deformation-type arguments and Lusternik–Schnirelmann theory, we prove that problem (1) admits infinitely many geometrically distinct solutions. © 2020 John Wiley & Sons, Ltd.

César T. Ledesma and Josias V. Baca were partially supported by CONCYTEC, Peru, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES.”

This work was partially supported by CONCYTEC PERU, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES”

Y. Ma is supported by Key R&D plan of Jiangxi Province, No:20181ACE50029, Y. Wang is supported by NNSF of China, No:12001252 and 12071189, by the Jiangxi Provincial Natural Science Foundation, No:20202ACBL201001 and 20202BAB201005, by the Science and Technology Research Project of Jiangxi Provincial Department of Education, No: 200325 and 200307, C. Torres was partially supported by CONCYTEC, Peru, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES”