Multiplicity of Positive Solutions for Kirchhoff Systems
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Multiplicity of Positive Solutions for Kirchhoff Systems Qing-Jun Lou1 Received: 3 June 2019 / Revised: 24 October 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The paper studies the combined effect of concave–convex nonlinearities on the number of solutions for a Kirchhoff system, which involves weight functions and critical exponents. Using Nehari manifold, the existence of multiple positive solutions is established when the pair of the parameters is under a certain range. Keywords Kirchhoff system · Critical Sobolev exponent · Nehari manifold · Positive solutions Mathematics Subject Classification 35J20 · 35J60
1 Introduction and Main Results The paper considers the existence and multiplicity of solutions for the following Kirchhoff-type system involving concave–convex nonlinearities ⎧ α 2 ⎪ ⎪ |u|α−2 u |v|β in , − a u = λ f (x) |u|q−2 u + + b |∇u| 1 1 ⎪ ⎪ α + β ⎪ ⎨ β 2 |u|α |v|β−2 v in , v = μg(x) |v|q−2 v + + b |∇v| − a ⎪ 2 2 ⎪ ⎪ α + β ⎪ ⎪ ⎩ u, v ∈ H01 (), (E λ,μ )
Communicated by Yong Zhou. This work is supported by the Doctoral Fund of University of Jinan (No. 160100414).
B 1
Qing-Jun Lou [email protected] School of Mathematical Sciences, University of Jinan, 250022 Jinan, People’s Republic of China
123
Q.-J. Lou
where ⊂ R N refers to a bounded domain with smooth boundary, α > 1, β > 1 satisfy α + β = 2∗ = 2N /(N − 2) (N = 3, 4) , 1 < q < 2, ai , bi > 0 (i = 1, 2), (λ, μ) ∈ (0, +∞) × (0, +∞). The weight functions f , g satisfy the following conditions: ¯ (F) f ∈ C(), f > 0. ¯ g > 0. (G) g ∈ C(),
Problem (E λ,μ ) is called nonlocal owning to the presence of b1 |∇u|2 and b2 |∇v|2 . In the previous literature, b( |∇u|2 )u appears in the Kirchhoff equation ⎧ ⎪ 2 ⎨ − a+b |∇u| u = f (x, u) ⎪ ⎩u = 0
in ,
(E 1 )
on ∂,
which is related to the stationary analogue of the equation u tt − a + b |∇u|2 u = f (x, u),
(E 2 )
where u denotes the displacement, f (x, t) is the external force, a refers to the initial tension and b is related to the intrinsic properties of the string. The equation was first proposed by Kirchhoff [22] as an extension of the classical D’Alembert’s wave equation to describe free vibrations of elastic strings. This model describes the changes in length of the string produced by transverse vibrations. However, equation (E 2 ) did not draw much attention until Lions [25] introduced an abstract framework to the problem [4,6]. Recently, many results about the existence and multiplicity of nontrivial solutions of equation (E 1 ) have been obtained, e.g., [8–11,16,17,23,24,28,32]. Moreover, the Kirchhoff equations with critical nonlinearities have been widely studied, e.g., [13– 15,18,19,27] and their references therein. In [7], using Nehari manifold, the authors studied equation (E 1 ), in which f (x, s) is replaced by λ f (x) |s|q−2 s + g (x) |s| p−2 s, where 1 < q < 2 < p < 2∗ , and they obtained the existence and multiplicity of solutions. For (E λ,μ ), when a = 0 and b = 1, it re
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