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Existence of positive solutions for nonlinear Kirchhoff type problems in R 3 with critical Sobolev exponent
Author(s) -
Li Gongbao,
Ye Hongyu
Publication year - 2013
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.3000
Subject(s) - mathematics , sobolev space , nonlinear system , exponent , type (biology) , critical exponent , function (biology) , mathematical analysis , extension (predicate logic) , sign (mathematics) , mountain pass theorem , sobolev inequality , pure mathematics , geometry , scaling , quantum mechanics , physics , biology , ecology , linguistics , philosophy , evolutionary biology , computer science , programming language
In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent:− a + b ∫R 3| Du | 2Δu + u = f ( x , u ) + u 5 , x ∈ R 3 ,u ∈ H 1(R 3) , u > 0 , x ∈ R 3 ,where a , b > 0 are constants. Under certain assumptions on the sign‐changing function f ( x , u ), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem−ϵ 2 a + ϵb ∫R 3| Du | 2Δu + V ( x ) u = f ( u ) , x ∈ R 3 ,u ∈ H 1(R 3) , u > 0 , x ∈ R 3 ,where ϵ > 0 is a parameter, V ( x ) is a positive continuous potential, and f ( u ) ∽ | u | p − 2 u with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.