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Archiv der Mathematik

Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials Zhipeng Yang

and Yuanyang Yu

Abstract. In this paper, we study the following nonlinear elliptic systems:  −Δu1 + V1 (x)u1 = ∂u1 F (x, u) x ∈ RN , x ∈ RN , −Δu2 + V2 (x)u2 = ∂u2 F (x, u) where u = (u1 , u2 ) : RN → R2 , F and Vi are periodic in x1 , . . . , xN and 0∈ / σ(− Δ + Vi ) for i = 1, 2, where σ(− Δ + Vi ) stands for the spectrum of the Schr¨ odinger operator − Δ + Vi . Under some suitable assumptions on F and Vi , we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009). Mathematics Subject Classification. 35J47, 35J50. Keywords. Nonlinear elliptic systems, Geometrically distinct solutions, Variational methods.

1. Introduction and main results. In this paper, we study the multiplicity of solutions for the nonlinear elliptic systems:  −Δu1 + V1 (x)u1 = f1 (x, u), x ∈ RN , (1.1) x ∈ RN , −Δu2 + V2 (x)u2 = f2 (x, u), where f := (f1 , f2 ) = ∂u F and F : RN × R2 → R. This type of systems arises when one considers standing wave solutions of time-dependent 2-coupled Schr¨ odinger systems of the form  1 i ∂φ ∂t = − Δφ1 + a1 (x)φ1 − g1 (x, |φ|)φ1 , (1.2) 2 i ∂φ ∂t = − Δφ2 + a2 (x)φ2 − g2 (x, |φ|)φ2 , where φ = (φ1 , φ2 ), i is the imaginary unit, ai (x) is a potential function, gi is a coupled nonlinear function modeling various types of the interaction effect

Z. Yang and Y. Yu

Arch. Math.

among many particles. System (1.2) has applications in many physical problems, especially in nonlinear optics and in Bose–Einstein condensates theory for multispecies Bose–Einstein condensates (see [1,10,14]). A standing wave solution of system (1.2) is a solution of the form φi (x, t) = e−iλi t ui (x), λi ∈ R, t > 0, and (u1 , u2 ) solves the system (1.1) with Vi (x) = ai (x)−λi , fi (x, u) = gi (x, |u|)ui for i = 1, 2. System (1.2) has been studied by some authors quite recently. In a bounded smooth domain Ω ⊂ RN , the similar systems were extensively studied by some authors, see for instance [3,5–7,19] and the references therein. The problem settled on the whole space RN was also considered recently in some works. One of the main difficulties of this problem is the lack of the compactness of the Sobolev embedding. And the second difficulty is that the negative definite space of the quadratic form which appears in the energy functional is infinitely dimensional, i.e., the energy functional is strongly indefinite. There are many different conditions and methods involved to avoid these difficulties, we refer to [4,8,9,16,24,25] and the references therein. Recall that the spectrum σ(− Δ + V ) of − Δ + V is purely continuous and may contain gaps, i.e., open intervals free of spectrum (see [18]). In [21], Szulkin and Weth considered the following Schr¨ odinger equation: − Δu + V (x)u = f (x, u) x ∈ RN ,

(1.3)

and proved that Eq. (1.3) possesses a ground state solution under the assumption