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Ground states for Kirchhoff‐type equations with critical or supercritical growth
Author(s) -
Li Quanqing,
Teng Kaimin,
Wu Xian
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.4485
Subject(s) - mathematics , supercritical fluid , ground state , type (biology) , nonlinear system , convergence (economics) , mathematical analysis , mathematical physics , physics , quantum mechanics , thermodynamics , ecology , economics , biology , economic growth
In this paper, we study the following Kirchhoff‐type equation with critical or supercritical growth− ( a + b ∫ R 3| ∇ u | 2 d x ) △ u + V ( x ) u = f ( x , u ) + λ | u | p − 2 u , x ∈ R 3 ,where a >0, b >0, λ >0, p ≥6 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x , we prove that the equation has a ground state solution for small λ >0 by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as b ↘0. Our main contribution is related to the fact that we are able to deal with the case p >6.