Existence of Positive Solutions for an Elliptic System Involving Nonlocal Operator

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DOI: 10.1007/s13226-020-0464-y

EXISTENCE OF POSITIVE SOLUTIONS FOR AN ELLIPTIC SYSTEM INVOLVING NONLOCAL OPERATOR Sh. Shahriyari∗ , H. Zahmatkesh∗ and S. Shakeri∗∗ ∗ Department

of Mathematics, Gorgan Branch,

Islamic Azad University, Gorgan, Iran ∗∗ Department

of Mathematics, Ayatollah Amoli Branch,

Islamic Azad University, Amol, Iran e-mails: [email protected], [email protected]; zahmatkesh [email protected], zahmatkesh [email protected]; [email protected], [email protected] (Received 30 October 2018; accepted 8 April 2019) Using the method of sub-super solutions and comparison principle, we study the existence of positive solutions for a class of quasilinear elliptic systems with sign-changing weights involving nonlocal operator. Key words : Kirchhoff-type systems; sub-super solutions; comparison principle; sign-changing weights. 2010 Mathematics Subject Classification : 35J60, 35J65.

1. I NTRODUCTION In this paper, we consider the existence of positive solutions for a class of quasilinear elliptic systems of the form

 ¡R ¢  −M1 Ω |∇u|p dx ∆p u = a(x) (λuα v γ + µf (u))    ¡R ¢ ¡ ¢ −M2 Ω |∇v|q dx ∆q v = b(x) λuδ v β + µg(v)     u=v=0

in Ω, in Ω,

(1.1)

on ∂Ω,

where Ω is a smooth bounded domain in RN , 1 < p, q < N , α, β, γ, δ are constants and λ, µ are positive parameters.

806

Sh. SHAHRIYARI, H. ZAHMATKESH AND S. SHAKERI Problems involving the p-Laplacian arise from many branches of pure mathematics as in the

theory of quasiregular and quasiconformal mapping (see [7]) as well as from various problems in mathematical physics notably the flow of non-Newtonian fluids. The structure of positive solutions for quasilinear reaction-diffusion systems (nonlinear Newtonian filtration systems) and semilinear reaction-diffusion systems (Newtonian filtration systems) is a front topic in the study of static electric fields in dielectric media, in which the potential is described by the boundary value problem of a static non-Newtonian filtration system, called the Poisson Boltzmann problem. This kind of problems also appears in the study of the non-Newtonian or Newtonian turbulent filtration in porous media and so on, which have extensive engineering background. In recent years, problems involving Kirchhoff type operators have been studied in many papers; we refer to [2, 4, 5, 6,13, 14, 15], in which the authors have used variational and topological methods to get the existence of solutions for (1.1). The main tool used in this study is the method of sub- and supersolutions. Our result in this note improves the previous one [8] in which M1 (t) = M2 (t) ≡ 1. We emphasize that it is really necessary to impose the boundedness of the Kirchhoff functions Mi , i = 1, 2. To our best knowledge, this is a new research topic for nonlocal problems; see [2, 3, 9]. In this paper, we denote by W01,r (Ω) (1 ≤ r < ∞) the completion of C0∞ (Ω), with respect to the norm

µZ kukr =

¶1 r . |∇u| dx r



Let us consider the following eigenvalue problem for the r-Laplace operator −∆r u, see [11]: ( −∆r u =