Infinitely many solutions for class of Neumann quasilinear elliptic systems

  • Davood Maghsoodi Shoorabi1Email author and

    Affiliated with

    • Ghasem Alizadeh Afrouzi2

      Affiliated with

      Boundary Value Problems20122012:54

      DOI: 10.1186/1687-2770-2012-54

      Received: 30 January 2012

      Accepted: 6 May 2012

      Published: 6 May 2012

      Abstract

      We investigate the existence of infinitely many weak solutions for a class of Neumann quasilinear elliptic systems driven by a (p1, ..., p n )-Laplacian operator. The technical approach is fully based on a recent three critical points theorem.

      AMS subject classification: 35J65; 34A15.

      Keywords

      infinitely many solutions Neumann system critical point theory variational methods

      1 Introduction

      The purpose of this article is to establish the existence of infinitely many weak solutions for the following Neumann quasilinear elliptic system
      - Δ p i u i + a i ( x ) u i p i - 2 u = λ F u i ( x , u 1 , , u n ) in Ω , u i ν = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ1_HTML.gif
      (1)

      for i = 1, ..., n, where Ω ⊂ ℝ N (N ≥ 1) is a non-empty bounded open set with a smooth boundary ∂Ω, p i > N for i = 1, ..., n, Δ p i u i = div ( u i p i - 2 u i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq1_HTML.gif is the p i -Laplacian operator, a i L (Ω) with ess infΩ a i > 0 for i = 1, ..., n, λ > 0, and F: Ω × ℝ n → ℝ is a function such that the mapping (t1, t2,..., t n ) → F (x, t1, t2,..., t n ) is in C1 in ℝ n for all x Ω , F t i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq2_HTML.gif is continuous in Ω × ℝ n for i = 1,..., n, and F (x, 0,..., 0) = 0 for all x ∈ Ω and ν is the outward unit normal to ∂Ω. Here, F t i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq3_HTML.gif denotes the partial derivative of F with respect to t i .

      Precisely, under appropriate hypotheses on the behavior of the nonlinear term F at infinity, the existence of an interval Λ such that, for each λ ∈ Λ, the system (1) admits a sequence of pairwise distinct weak solutions is proved; (see Theorem 3.1). We use a variational argument due to Ricceri which provides certain alternatives in order to find sequences of distinct critical points of parameter-depending functionals. We emphasize that no symmetry assumption is required on the nonlinear term F (thus, the symmetry version of the Mountain Pass theorem cannot be applied). Instead of such a symmetry, we assume a suitable oscillatory behavior at infinity on the function F.

      We recall that a weak solution of the system (1) is any u = u 1 , . . . , u n W 1 , p 1 Ω × . . . × W 1 , p n Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq4_HTML.gif such that
      Ω i = 1 n u i ( x ) p i - 2 u i ( x ) v i ( x ) + a i ( x ) u i ( x ) p i - 2 u i ( x ) v i ( x ) d x - λ Ω i = 1 n F u i ( x , u 1 ( x ) , . . . u n ( x ) ) v i ( x ) d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equa_HTML.gif

      for all v = v 1 , . . . , v n W 1 , p 1 Ω × . . . × W 1 , p n Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq5_HTML.gif

      For a discussion about the existence of infinitely many solutions for differential equations, using Ricceri's variational principle [1]and its variants [2, 3] we refer the reader to the articles [416].

      For other basic definitions and notations we refer the reader to the articles [1722]. Here, our motivation comes from the recent article [8]. We point out that strategy of the proof of the main result and Example 3.1 are strictly related to the results and example contained in [8].

      2 Preliminaries

      Our main tool to ensure the existence of infinitely many classical solutions for Dirichlet quasilinear two-point boundary value systems is the celebrated Ricceri's variational principle [[1], Theorem 2.5] that we now recall as follows:

      Theorem 2.1. Let X be a reflexive real Banach space, let Φ, Ψ: X → ℝ be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf X Φ, let us put
      φ ( r ) : = inf u Φ - 1 - , r sup v Φ - 1 - , r Ψ ( v ) - Ψ ( u ) r - Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equb_HTML.gif
      and
      γ : = lim inf r + φ ( r ) , δ : = lim inf r ( inf X Φ ) + φ ( r ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equc_HTML.gif

      Then, one has

      (a) for every r > inf X Φ and every λ 0 , 1 φ ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq6_HTML.gif, the restriction of the functional I λ = Φ - λ Ψ to Φ-1(] - ∞, r[) admits a global minimum, which is a critical point (local minimum) of I λ in X.

      (b) If γ < +∞ then, for each λ 0 , 1 γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq7_HTML.gif, the following alternative holds:

      either

      (b1) I λ possesses a global minimum,

      or

      (b2) there is a sequence {u n } of critical points (local minima) of I λ such that
      lim n + Φ ( u n ) = + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equd_HTML.gif

      (c) If δ < +∞ then, for each λ 0 , 1 δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq8_HTML.gif, the following alternative holds:

      either

      (c1) there is a global minimum of Φ which is a local minimum of I λ ,

      or

      (c2) there is a sequence {u n } of pairwise distinct critical points (local minima) of I λ that converges weakly to a global minimum of Φ.

      We let X be the Cartesian product of n Sobolev spaces W 1 , p 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq9_HTML.gif, W 1 , p 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq10_HTML.gif,... and W 1 , p n ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq11_HTML.gif, i.e., X = i = 1 n W 1 , p i ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq12_HTML.gif, equipped with the norm
      u 1 , u 2 , , u n = i = 1 n u i p i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Eque_HTML.gif
      where
      u i p i = Ω u i ( x ) p i + a i ( x ) u i ( x ) p i d x 1 p i , i = 1 , , n . C = max sup u i W 1 , p i ( Ω ) \ { 0 } sup x Ω u ( x ) p i u i p i p i ; i = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ2_HTML.gif
      (2)
      Since p i > N for 1 ≤ i ≤ n, one has C < +∞. In addition, if Ω is convex, it is known [23] that
      sup u i W 1 , p i ( Ω ) \ { 0 } sup x Ω u i ( x ) u i p i 2 p i - 1 p i max 1 a i 1 1 p i ; diam ( Ω ) N 1 p i p i - 1 p i - N m ( Ω ) p i - 1 p i a i a i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equf_HTML.gif

      for 1 ≤ i ≤ n, where ||·||1 = ∫Ω|·(x)| dx, ||·|| = supx∈Ω|·(x)| and m(Ω) is the Lebesgue measure of the set Ω, and equality occurs when Ω is a ball.

      In the sequel, let p ¯ = min { p i ; 1 i n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq13_HTML.gif.

      For all γ > 0 we define
      K ( γ ) = ( t 1 , , t n ) n : i = 1 n t i γ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ3_HTML.gif
      (3)

      3 Main results

      We state our main result as follows:

      Theorem 3.1. Assume that

      (A1)
      lim inf ξ + Ω sup ( t 1 , . . . , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < i = 1 n ( p i C ) 1 p i p - lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equg_HTML.gif

      where K ( ξ ) = { ( t 1 , , t n ) | i = 1 n t i ξ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq14_HTML.gif (see (3)).

      Then, for each
      λ Λ : = 1 lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n | | a i | | 1 | t i | p i , i = 1 n ( p i C ) 1 p i p - lim inf ξ + Ω sup ( t 1 , . . . , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equh_HTML.gif

      the system (1) has an unbounded sequence of weak solutions in X.

      Proof. Define the functionals Φ, Ψ: X → ℝ for each u = (u1, ..., u n ) ∈ X, as follows
      Φ ( u ) = i = 1 n u i p i p i p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equi_HTML.gif
      and
      Ψ ( u ) = Ω F ( x , u 1 ( x ) , , u n ( x ) ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equj_HTML.gif
      It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative at the point uX is the functional Ψ'(u) ∈ X*, given by
      Ψ ( u ) ( v ) = Ω i = 1 n F u i ( x , u 1 ( x ) , , u n ( x ) ) v i ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equk_HTML.gif
      for every v = (v1, ..., v n ) ∈ X, and Ψ': XX* is a compact operator. Moreover, Φ is a sequentially weakly lower semicontinuous and Gâteaux differentiable functional whose Gâteaux derivative at the point uX is the functional Φ' (u) ∈ X*, given by
      Φ ( u 1 , , u n ) ( v 1 , , v n ) Ω i = 1 n u i ( x ) p i - 2 u i ( x ) v i ( x ) + a i ( x ) u i ( x ) p i - 2 u i ( x ) v i ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equl_HTML.gif

      for every v = (v1, ..., v n ) ∈ X. Furthermore, (Φ')-1: X*X exists and is continuous.

      Put I λ : = Φ - λ Ψ. Clearly, the weak solutions of the system (1) are exactly the solutions of the equation I λ ( u 1 , , u n ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq15_HTML.gif. Now, we want to show that
      γ < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equm_HTML.gif
      Let {ξ m } be a real sequence such that ξ m → +∞ as m → ∞ and
      lim m Ω sup ( t 1 , , t n ) K ( ξ m ) F ( x , t 1 , , t n ) d x ξ m p - = lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equn_HTML.gif
      Put r m = ξ m p - i = 1 n ( p i C ) 1 p i p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq16_HTML.gif for all m ∈ ℕ. Since
      sup x Ω u i ( x ) p i C u i p i p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equo_HTML.gif
      for each u i W 1 , p i ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq17_HTML.gif for 1 ≤ in, we have
      sup x Ω i = 1 n u i ( x ) p i p i C i = 1 n u i p i p i p i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ4_HTML.gif
      (4)
      for each u = (u1, u2, ..., u n ) ∈ X. This, for each r > 0, together with (4), ensures that
      Φ - 1 - , r u X ; sup i = 1 n u i ( x ) p i p i C r for each x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equp_HTML.gif
      Hence, an easy computation shows that i = 1 n u i ξ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq18_HTML.gif whenever u = (u1, ..., u n ) ∈ Φ-1(] - ∞, r m ]). Hence, one has
      φ ( r m ) = inf u Φ - 1 - , r m ( sup v Φ - 1 - , r m Ψ ( v ) ) - Φ ( u ) r m - Φ ( u ) sup v Φ - 1 - , r m Ψ ( v ) r m Ω sup ( t 1 , , t n ) K ( ξ m ) F ( x , t 1 , , t n ) d x ξ m p - i = 1 n ( p i C ) 1 p i p - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equq_HTML.gif
      Therefore, since from Assumption (A1) one has
      lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equr_HTML.gif
      we deduce
      γ lim inf m + φ ( r m ) i = 1 n ( p i C ) 1 p i p - lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ5_HTML.gif
      (5)
      Assumption (A1) along with (5), implies
      Λ 0 , 1 γ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equs_HTML.gif

      Fix λ ∈ Λ. The inequality (5) concludes that the condition (b) of Theorem 2.1 can be applied and either I λ has a global minimum or there exists a sequence {u m } where u m = (u1m, ..., u nm ) of weak solutions of the system (1) such that limm→∞||(u1m, ..., u nm )|| = +.

      Now fix λ ∈ Λ and let us verify that the functional I λ is unbounded from below. Arguing as in [8], consider n positive real sequences { d i , m } i = 1 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq19_HTML.gif such that i = 1 n d i , m 2 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq20_HTML.gif as m

      and
      lim m + Ω F ( x , d 1 , m , , d n , m ) d x i = 1 n d i , m p i p i = lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ6_HTML.gif
      (6)
      For all m ∈ ℕ define w m (x) = (d1, m, ..., dn, m). For any fixed m ∈ ℕ, w m X and, in particular, one has
      Φ ( w m ) = i = 1 n d i , m p i a i 1 p i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equt_HTML.gif
      Then, for all m ∈ ℕ,
      I λ ( w m ) = Φ ( w m ) - λ Ψ ( w m ) = i = 1 n d i , m p i a i 1 p i - λ Ω F ( x , d 1 , m , , d n , m ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equu_HTML.gif
      Now, if
      lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 | t i | p i p i < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equv_HTML.gif
      we fix ε 1 λ lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq21_HTML.gif. From (6) there exists τ ε such that
      Ω F ( x , d 1 , m , , d n , m ) d x > ε lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i i = 1 n d i , m p i a i 1 p i m > τ ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equw_HTML.gif
      therefore
      I λ ( w m ) 1 - λ ε lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i i = 1 n d i , m p i a i 1 p i m > τ ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equx_HTML.gif
      and by the choice of ε, one has
      lim m + [ Φ ( w m ) - λ Ψ ( w m ) ] = - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equy_HTML.gif
      If
      lim sup ξ + Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equz_HTML.gif
      let us consider K > 1 λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq22_HTML.gif. From (6) there exists τ K such that
      Ω F ( x , d 1 , m , , d n , m ) d x > K i = 1 n d i , m p i a i 1 p i m > τ K , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equaa_HTML.gif
      therefore
      I λ ( w m ) ( 1 - λ K ) i = 1 n d i , m p i a i 1 p i m > τ K , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equab_HTML.gif
      and by the choice of K, one has
      lim m + [ Φ ( w m ) - λ Ψ ( w m ) ] = - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equac_HTML.gif
      Hence, our claim is proved. Since all assumptions of Theorem 2.1 are satisfied, the functional I λ admits a sequence {u m = (u1m, ..., u nm )} ⊂ X of critical points such that
      lim m ( u 1 m , , u n m ) = + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equad_HTML.gif

      and we have the conclusion.   □

      Here, we give a consequence of Theorem 3.1.

      Corollary 3.2. Assume that

      (A2) lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < i = 1 n ( p i C ) 1 p i p - ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq23_HTML.gif

      (A3) lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i > 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq24_HTML.gif

      Then, the system
      - Δ p i u i + a i ( x ) u i p i - 2 u = F u i ( x , u 1 , , u n ) i n Ω , u i ν = 0 o n Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equae_HTML.gif

      for 1 ≤ i ≤ n, has an unbounded sequence of classical solutions in X.

      Now, we want to present the analogous version of the main result (Theorem 3.1) in the autonomous case.

      Theorem 3.3. Assume that

      (A4)
      lim inf ξ + sup ( t 1 , , t n ) K ( ξ ) F ( t 1 , , t n ) ξ p - < i = 1 n ( p i C ) 1 p i p - lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n F ( t 1 , , t n ) i = 1 n a i 1 t i p i p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equaf_HTML.gif

      where K ( ξ ) = { ( t 1 , , t n ) | i = 1 n t i ξ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_IEq25_HTML.gif (see (3)).

      Then, for each
      λ Λ : = 1 F ( t 1 , , t n ) lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n i = 1 n a i 1 t i p i , i = 1 n ( p i C ) 1 p i p - lim inf ξ + sup ( t 1 , , t n ) K ( ξ ) F ( t 1 , , t n ) ξ p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equag_HTML.gif
      the system
      - Δ p i u i + a i ( x ) u i p i - 2 u = λ F u i ( u 1 , , u n ) i n Ω , u i ν = 0 o n Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equah_HTML.gif

      has an unbounded sequence of weak solutions in X.

      Proof. Set F (x, u1, ..., u n ) = F (u1, ..., u n ) for all x ∈ Ω and (u1, ..., u n ) ∈ ℝ n . The conclusion follows from Theorem 3.1. □

      Remark 3.1. We observe in Theorem 3.1 we can replace ξ → +∞ and (t1, ..., t n ) → (+, ..., +∞) with ξ → 0+ (t1, ..., t n ) → (0+, ..., 0+), respectively, that by the same way as in the proof of Theorem 3.1 but using conclusion (c) of Theorem 2.1 instead of (b), the system (1) has a sequence of weak solutions, which strongly converges to 0 in X.

      Finally, we give an example to illustrate the result.

      Example 3.1. Let Ω ⊂ ℝ2 be a non-empty bounded open set with a smooth boundary ϑΩ and consider the increasing sequence of positive real numbers given by
      a n : = 2 , a n + 1 : = n ! ( a n ) 5 4 + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equai_HTML.gif
      for every n ≥ 1. Define the function
      F ( t 1 , t 2 ) = ( a n + 1 ) 5 e - 1 1 - [ ( t 1 - a n + 1 ) 2 + ( t 2 - a n + 1 ) 2 ] ( t 1 , t 2 ) n 1 B ( ( a n + 1 , a n + 1 ) , 1 ) , 0 otherwise http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equ7_HTML.gif
      (7)

      where B((an+1, an+1), 1)) be the open unit ball of center (an+1, an+1). We observe that the function F is non-negative, F (0, 0) = 0, and FC1(ℝ2). We will denote by f and g, respectively, the partial derivative of F respect to t1 and t2. For every n ∈ ℕ, the restriction F on B((an+1, an+1), 1) attains its maximum in (an+1, an+1) and F (an+1, an+1) = (an+1)5,

      then
      lim sup n + F ( a n + 1 , a n + 1 ) a n + 1 3 3 + a n + 1 4 4 = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equaj_HTML.gif
      So
      lim sup ( t 1 , t 2 ) ( + , + ) F ( t 1 , t 2 ) t 1 3 3 + t 2 4 4 = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equak_HTML.gif
      On the other by setting y n = an+1- 1 for every n ∈ ℕ, one has
      sup ( t 1 , t 2 ) K ( y n ) F ( t 1 , t 2 ) = a n 5 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equal_HTML.gif
      Then
      lim n sup ( t 1 , t 2 ) K ( y n ) F ( t 1 , t 2 ) ( a n + 1 - 1 ) 3 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equam_HTML.gif
      and hence
      lim inf ξ sup ( t 1 , t 2 ) K ( ξ ) F ( t 1 , t 2 ) ξ 3 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equan_HTML.gif
      Finally
      0 = lim inf ξ + sup ( t 1 , t 2 ) K ( ξ ) F ( t 1 , t 2 ) ξ 3 < ( ( 3 C ) 1 3 + ( 4 C ) 1 4 ) 3 lim sup ( t 1 , t 2 ) ( + , + ) ( t 1 , t 2 ) + n F ( t 1 , t 2 ) t 1 3 3 + t 2 4 4 = + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equao_HTML.gif
      So, since all assumptions of Theorem 3.3 is applicable to the system
      - Δ 3 u + u u = λ f ( u , v ) in Ω , - Δ 4 v + v 2 g = λ g ( u , v ) in Ω , u ν = v ν = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-54/MediaObjects/13661_2012_Article_144_Equap_HTML.gif

      for every λ ∈ [0, +[.

      Declarations

      Authors’ Affiliations

      (1)
      Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU)
      (2)
      Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran

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      © Shoorabi and Afrouzi; licensee Springer. 2012

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