Báo cáo toán học: On a System of Semilinear Elliptic Equations on an Unbounded Domain

Trong bài báo này chúng ta nghiên cứu sự tồn tại của các giải pháp yếu của vấn đề Dirichlet cho một hệ thống phương trình elliptic semilinear trên một miền không bị chặn trong Rn.
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Báo cáo toán học: "On a System of Semilinear Elliptic Equations on an Unbounded Domain" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:4 (2005) 381–389 RI 0$7+(0$7,&6 ‹ 9$67 On a System of Semilinear Elliptic Equations on an Unbounded Domain Hoang Quoc Toan Faculty of Math., Mech. and Inform. Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam Received May 12, 2004 Revised August 28, 2005Abstract. In this paper we study the existence of weak solutions of the Dirichletproblem for a system of semilinear elliptic equations on an unbounded domain in Rn .The proof is based on a fixed point theorem in Banach spaces.1. IntroductionIn the present paper we consider the following Dirichlet problem: −Δu + q (x)u = αu + βv + f1 (u, v ) in Ω (1.1) −Δv + q (x)v = Δu + γv + f2 (u, v ) u|∂ Ω = 0, v |∂ Ω = 0 u(x) → 0, v (x) → 0 as |x| → +∞ (1.2) nwhere Ω is a unbounded domain with smooth boundary ∂ Ω in a R , α, β, δ, γ aregiven real numbers, β > 0, δ > 0; q (x) is a function defined in Ω, f1 (u, v ), f2 (u, v )are nonlinear functions for u, v such that q (x) ∈ C 0 (R), and ∃q0 > 0, q (x) ≥ q0 , ∀x ∈ Ω (1.3) q (x) → +∞ as |x| → +∞fi (u, v ) are Lipschitz continuous in Rn with constants ki (i = 1, 2) ∀(u, v ), (¯, v ) ∈ R2 . |fi (u, v ) − fi (¯, v )| ki (|u − u| + |v − v |), u¯ ¯ ¯ u¯ (1.4)382 Hoang Quoc Toan The aim of this paper is to study the existence of weak solution of theproblem (1.1)-(1.2) under hypothesis (1.3), (1.4) and suitable conditions for theparameters α, β, δ, γ. We firstly notice that the problem Dirichlet for the system (1.1) in a boundedsmooth domain have been studied by Zuluaga in [6]. Throughout the paper, (., .) and . denotes the usual scalar product and ◦the norm in L2 (Ω); H 1 (Ω), H 1 (Ω) are the usual Sobolev’s spaces.2. Preliminaries and Notations ∞We define in C0 (Ω) the norm (as in [1]) 1 2 |Du|2 + qu2 dx ∞ , ∀u ∈ C0 (Ω) u = (2.1) q ,Ω Ωand the scalar product aq (u, v ) = (u, v )q = (DuDv + qu.v )dx (2.2) Ω ∂ u ∂u ∂u ∞ ,··· , , ∀u, v ∈ C0 (Ω). where Du = , ∂x1 ∂x2 ∂xnThen we introduce the space Vq0 (Ω) defined as the completion of C0 (Ω) with ∞ 0respect to the norm . q,Ω . Furthermore, the space Vq (Ω) can be considered asa Sobolev-Slobodeski’s space with weight.Proposition 2.1. (see [1]) Vq0 (Ω) is a Hilbert space which is dense in L2 (Ω),and the embedding of Vq0 (Ω) into L2 (Ω) is continuous and compact. We define by the Lax-Milgram lemma a unique operator Hq in L2 (Ω) suchthat (Hq u, v ) = aq (u, v ), ∀u ∈ D(Hq ), ∀v ∈ Vq0 (Ω)where D(Hq ) = {u ∈ Vq0 (Ω) : Hq u = (−Δ + q )u ∈ L2 (Ω)}. It is obvious that the operator Hq : D(Hq ) ⊂ L2 (Ω) → L2 (Ω)is a linear operator with range R(Hq ) ⊂ L2 (Ω). Since q (x) is positive, the operator Hq is positive in the sense that: (Hq u, u)L2 (Ω) ≥ 0, ∀u ∈ D(Hq )and selfadjoint (Hq u, v )L2 (Ω) = (u, Hq v )L2 (Ω) , ∀u, v ∈ D(Hq ). Hq 1 − is defined on R(Hq ) ∩ L2 (Ω) with range D(Hq ), considered asIts inversean operator into L2 (Ω). By Proposition 2.1 it follows that Hq 1 is a compact −System of Semilinear Elliptic Equations on an Unbounded Domain 383operator in L2 (Ω). Hence the spectrum of Hq consists of a countable sequenceof eigenvalues {λk }∞ , each with finite multiplicity and the first eigenvalue λ1 k=1is isolated and simple: ··· λk · · · , λk → +∞ as k → +∞. 0 < λ1 < λ2Every eigenfunction ϕk (x) associated with λk (k = 1, 2, · · · ) is continuous andbounded on Ω and there exist positive constants α and β such that αe−β |x| |ϕk (x)| for |x| large enough.Moreover eigenfunction ϕ1 (x) > 0 in Ω (see [1]).Proposition 2.2. (Maximum principle. see [1]) Assume that q (x) satisfies thehypothesis (1.3), and λ < λ1 . Then for any g (x) in L2 (Ω), there exists a uniquesolution u(x) of the following problem: ...