Ground state solutions for a class of nonlocal regional Schrödinger equation with nonperiodic potentials

In this article, we deal with the nonlinear Schrödinger equation with nonlocal regional diffusion 0.1( − Δ ) ρ α u + V ( x ) u = f ( x , u ) inℝ N , u ∈ H α ( ℝ N ) , where 0 <  α  < 1 , n  ≥ 2 , and f : ℝ N × ℝ → ℝ is a continuous function. The operator( − Δ ) ρ αis a variational version of the nonlocal regional Laplacian defined as∫ℝ N( − Δ ) ρ α u ( x ) φ ( x ) d x = ∫ℝ N∫ B ( 0 , ρ ( x ) )[ u ( x + z ) − u ( x ) ] [ φ ( x + z ) − φ ( x ) ] | z | N + 2 αd z d x , where ρ ∈ C ( ℝ N , ℝ + ) be a positive function. Considering that ρ ,  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.