Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent

In this paper we investigate the following Kirchhoff type elliptic boundary value problem involving a critical nonlinearity: - ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = μ g ( x , u ) + u 5 , u > 0 in Ω , u = 0 on ∂ Ω , ( K 1 ) here Ω ⊂ R 3 is a bounded domain with smooth boundary ∂ Ω , a , b ≥ 0 and a +...

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Bibliographic Details
Published in:Nonlinear differential equations and applications 2014-12, Vol.21 (6), p.885-914
Main Author: Naimen, Daisuke
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary:In this paper we investigate the following Kirchhoff type elliptic boundary value problem involving a critical nonlinearity: - ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = μ g ( x , u ) + u 5 , u > 0 in Ω , u = 0 on ∂ Ω , ( K 1 ) here Ω ⊂ R 3 is a bounded domain with smooth boundary ∂ Ω , a , b ≥ 0 and a + b > 0. Under several conditions on g ∈ C ( Ω ¯ × R , R ) and μ ∈ R , we prove the existence and nonexistence of solutions of (K1). This is some extension of a part of Brezis–Nirenberg’s result in 1983 .
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-014-0271-4