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Ground state solutions for the nonlinear Schrödinger‐Poisson systems with sum of periodic and vanishing potentials
Author(s) -
Xie Weihong,
Chen Haibo,
Shi Hongxia
Publication year - 2017
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.4602
Subject(s) - nehari manifold , mathematics , ground state , monotonic function , schrödinger's cat , poisson distribution , nonlinear system , mathematical physics , mathematical analysis , energy functional , state (computer science) , energy (signal processing) , sign (mathematics) , energy method , type (biology) , manifold (fluid mechanics) , pure mathematics , quantum mechanics , physics , mechanical engineering , ecology , statistics , algorithm , engineering , biology
We study the existence of ground state solutions for the following Schrödinger‐Poisson equations:− Δ u + V ( x ) u + ϕ u = f ( x , u ) ,inR 3 ,− Δ ϕ = u 2 , inR 3 ,where V = V p + V l o c ∈ L ∞ ( R 3 ) is the sum of a periodic potential V p and a localized potential V l o c and f satisfies the subcritical or critical growth. Although the Nehari‐type monotonicity assumption on f is not satisfied in the subcritical case, we obtain the existence of a ground state solution as a minimizer of the energy functional on Nehari manifold. Moreover, we show that the existence and nonexistence of ground state solutions are dependent on the sign of V l o c .