Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents

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.