Premium
Ground state solutions for a class of nonlocal regional Schrödinger equation with nonperiodic potentials
Author(s) -
Ledesma César T.,
Gutiérrez Hernán C.
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7005
Subject(s) - mathematics , nehari manifold , laplace operator , ground state , infinity , operator (biology) , class (philosophy) , function (biology) , type (biology) , mathematical analysis , state (computer science) , nonlinear system , manifold (fluid mechanics) , pure mathematics , mathematical physics , physics , quantum mechanics , mechanical engineering , ecology , biochemistry , chemistry , repressor , artificial intelligence , evolutionary biology , algorithm , biology , computer science , transcription factor , engineering , gene
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.