Existence and Multiplicity of Solutions for a Coupled System of Kirchhoff Type Equations
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS∗ Yaghoub JALILIAN Department of Mathematics, Razi University, Kermanshah, Iran E-mail : [email protected] Abstract
In this paper, we study the coupled system of Kirchhoff type equations Z 2α 2 |u|α−2 u|v|β , x ∈ R3 , |∇u| dx − a + b ∆u + u = α +β 3 R Z 2β 2 |∇v| dx ∆v + v = − a + b |u|α |v|β−2 v, x ∈ R3 , α + β 3 R u, v ∈ H 1 (R3 ),
where a, b > 0, α, β > 1 and 3 < α + β < 6. We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity. Also, using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method, we obtain the existence of infinitely many geometrically distinct solutions in the case when α, β ≥ 2 and 4 ≤ α + β < 6. Key words
Kirchhoff equation; Nehari-Poho˘zave manifold; constrained minimization; ground state solution
2010 MR Subject Classification
1
35J50; 35J10
Introduction
In this article, we investigate the existence and multiplicity of solutions to the coupled system of Kirchhoff type equations Z 2α 2 − a + b |∇u| dx ∆u + u = |u|α−2 u|v|β , x ∈ R3 , α + β 3 R Z 2β (1.1) 2 − a + b |∇v| dx ∆v + v = |u|α |v|β−2 v, x ∈ R3 , α + β 3 R u, v ∈ H 1 (R3 ), where a, b > 0, α, β > 1 and 3 < α + β < 6. Kirchhoff type equations are related to the stationary analogue of the equation Z utt − a + b |∇u|2 dx ∆u + V (x)u = g(x, u), R3
∗ Received
June 11, 2019; revised October 18, 2019.
(1.2)
1832
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
proposed by Kirchhoff [1] which is an extension of the classical D’Alembert’s wave equation for free transversal vibrations of a clamped string. Equation (1.2) arises in various physical and biological systems. For example, in biological systems, u could describe the population density in a biological phenomena such as bacteria spreading (see [2]). Motivated by the physical background and the mathematical difficulty caused by the term 2 R3 |∇u| dx, many researchers studied Kirchhoff type equations of the form
R
Z − a+b
RN
|∇u|2 dx ∆u + V (x)u = g(x, u),
x ∈ RN .
(1.3)
To see some recent contributions on the existence and multiplicity of solutions for equation (1.3), we refer the reader to [1, 3–14]. It is worth to mention that the problem will be more complicated when N = 3 and g(x, u) is not 4-superlinear at infinity (i.e., lim G(x,u) < ∞ where u4 u→∞ R G(x, u) = R3 g(x, s)ds). The difficulty is to get a bounded Palais-Smale sequence. Li and Ye in [11] obtained the first existence result for (1.3) in R3 with g(x, u) = |u|p−2 u and 3 < p < 6. They used the idea of [13] and introduced a new manifold constructed by the Nehari manifold and the Poho˘zave identity (Nehari-Poho˘zave manifold). Then by the constrained minimization and the concentration compactness principle, they proved the existence of a posi
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