Premium
Fractional Kirchhoff problem with critical indefinite nonlinearity
Author(s) -
do Ó João Marcos,
He Xiaoming,
Mishra Pawan Kumar
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800044
Subject(s) - mathematics , sublinear function , nehari manifold , sobolev space , bounded function , multiplicity (mathematics) , mathematical analysis , nonlinear system , critical exponent , domain (mathematical analysis) , pure mathematics , scaling , geometry , physics , quantum mechanics
We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the formM∫ ∫R 2 N| u ( x ) − u ( y ) | 2| x − y | N + 2 sd x d y( − Δ ) s u= λ f ( x ) | u | q − 2 u + | u |2 s ∗ − 2 u in Ω , u = 0 inR N ∖ Ω ,where Ω ⊂ R Nis a smooth bounded domain, 0 < s < 1 and 1 < q < 2 . Here M is the Kirchhoff coefficient and2 s ∗ = 2 N / ( N − 2 s )is the fractional critical Sobolev exponent. The parameter λ is positive and the f ( x ) is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.