Existence and concentration behavior of positive solutions for a Kirchhoff equation in R 3

We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem { − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , u > 0 in R 3 , where ε > 0 is a parameter and a , b > 0 are constants; V is...

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Bibliographic Details
Published in:Journal of Differential Equations 2012, Vol.252 (2), p.1813-1834
Main Authors: He, Xiaoming, Zou, Wenming
Format: Article
Language:English
Subjects:
Kirchhoff type equation
Positive solutions
Variational methods
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Summary:We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem { − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , u > 0 in R 3 , where ε > 0 is a parameter and a , b > 0 are constants; V is a positive continuous potential satisfying some conditions and f is a subcritical nonlinear term. We relate the number of solutions with the topology of the set where V attains its minimum. The results are proved by using the variational methods.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2011.08.035