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Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension
Author(s) -
Yao Xianzhong,
Mu Chunlai
Publication year - 2016
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.3821
Subject(s) - mathematics , nehari manifold , multiplicity (mathematics) , bounded function , sobolev space , domain (mathematical analysis) , mathematical analysis , dimension (graph theory) , type (biology) , boundary value problem , pure mathematics , nonlinear system , ecology , physics , quantum mechanics , biology
In this paper, we study the following Kirchhoff‐type elliptic problem− ( a + b ∫ Ω | ∇ u | 2 dx ) Δ u = λ u q − 1 + μ u2 ∗ − 1 , u > 0in Ω ,u = 0 ,on ∂ Ω ,where Ω ⊂ R N ( N ≥ 4 ) is a bounded domain with smooth boundary ∂ Ω, a , b , λ , μ > 0 and 1 < q < 2 ∗ =2 N /( N − 2). When N = 4, we obtain that there is a ground state solution to the problem for q ∈(2,4) by using a variational methods constrained on the Nehari manifold and also show the problem possesses infinitely many negative energy solutions for q ∈(1,2) by applying usual Krasnoselskii genus theory. In addition, we admit that there is a positive solution to the equations for N ≥5 under some suitable conditions. Copyright © 2016 John Wiley & Sons, Ltd.