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Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents
Author(s) -
Wang Zhuangzhuang,
Zeng Xiaoyu,
Zhang Yimin
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - Bosnian
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6256
Subject(s) - mathematics , sobolev space , critical exponent , scaling , algebraic number , mathematical analysis , elliptic curve , algebraic equation , nonlinear system , geometry , physics , quantum mechanics
We study the following Kirchhoff equation:− a ε 2 + b ε 4 − N∫R N| ∇ u | 2△ u = g ( x , u ) , a , b > 0 , N ≥ 3 , ( K ) where g ( x , u ) = K ( x ) | u |2 ∗ − 2 u or g ( x , u ) = − V ( x ) u + | u | p − 2 u ( 2 < p < 2 ∗ ) is critical or subcritical growth. By introducing suitable scalings, ( K ) is transformed into a semilinear elliptic equation coupled with an algebraic equation of the scaling parameter. Upon which, we prove the existence of multi‐peak solutions of ( K ) concentrating around the critical points of K ( x ) or V ( x ) . Moreover, we prove that ( K ) has only trivial solution if N ≥ 4 and b > 0 is suitably large.