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Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth
Author(s) -
Li Quanqing,
Teng Kaimin,
Wu Xian,
Wang Wenbo
Publication year - 2019
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.5441
Subject(s) - mathematics , fractional laplacian , supercritical fluid , mathematical physics , order (exchange) , function (biology) , laplace operator , mathematical analysis , physics , thermodynamics , finance , evolutionary biology , biology , economics
In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth( − Δ ) s u + V ( x ) u = f ( x , u ) + λ | u | p − 2 u , x ∈ R N , where 0 < s < 1, N > 2 s , λ > 0,2 s ∗ = 2 N N − 2 s , p ≥ 2 s ∗ , ( − Δ) s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case p > 2 s ∗ .