Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

and Applied Analysis 3 (H3) There exists r > 0, λ ∈ (λ 1 , λ 2 ) and μ ∈ (μ 1 , μ 2 ) such that M 1 λ 1 < m 1 λ, M 2 μ 1 < m 2 μ, and |u|, |V| ≤ r implies

A typical example for   is given by   () = || −(+2  ) ( = 1, 2).In this case L   is the fractional Laplace operator −(−Δ)   , where −(−Δ)   is defined by here   ∈ (0, 1) and  > 2  ( = 1, 2).The fractional Laplacian −(−Δ)   is a classical linear integrodifferential operator of order 2  which gives the standard Laplacian when   = 1 (see [1]).Denote by   the linear space of Lebesgue measurable functions  : R  → R such that the map (, )  ( () −  ()) 2   ( − ) where The space   is endowed with the norm The space   denotes the closure of  ∞ 0 (Ω) in   .By Lemmas 6 and 7 in [2], the space   is a Hilbert space which can be endowed with the norm defined as Since  = 0 a.e. in R  \ Ω, we have that the integral in ( 8) and ( 9) can be extended to all R 2 .Let  =  1 ×  2 be the Cartesian product of two Hilbert spaces, which is a reflexive Banach space endowed with the norm Denote by 0 <  1 <  2 ≤ ⋅ ⋅ ⋅ ≤   ≤ ⋅ ⋅ ⋅ the eigenvalues of the following nonlocal operator eigenvalue problem: Similarly, denote by 0 <  1 <  2 ≤ ⋅ ⋅ ⋅ ≤   ≤ ⋅⋅⋅ the eigenvalues of the following nonlocal operator eigenvalue problem: We say that (, V) ∈  is a weak solution of system (1) if, for every (, ) ∈ , one has The fractional Laplacian and nonlocal operators of elliptic type arise in both pure mathematical research and concrete applications, since these operators occur in a quite natural way in many different contexts.For an elementary introduction to this topic, see [2] and the references therein.Recently, some elliptic boundary problems driven by the nonlocal integrodifferential operator L  have been studied in the works [3][4][5][6][7][8].
Recently, problems involving Kirchhoff type operators have been studied in many papers; we refer to [9][10][11][12][13] in which the authors have used the variational method and topological method to get the existence of solutions.
In this paper, motivated by the above mentioned works, we will use Morse theory to investigate the multiplicity of solutions of problem (1).To the best of our knowledge, there is no effort being made in the literature to study the existence of solutions for problem (1).This paper will make some contribution to this research field.
In order to establish solutions for problem (1), we make the following assumptions.
The main result of this paper is as follows.

Preliminaries
For each (, V) ∈ , we define the functional J :  → R as follows: where It is easy to check that (, V) is a weak solution of problem (1) which is equivalent to being a critical point of the functional J.
First let us recall the definition of the local linking which plays an important role in our paper.Definition 2. Let  be a Banach space with a direct sum decomposition  =  1 ⊕  2 .The functional  ∈  1 (, R) has a local linking at 0 with respect to ( 1 ,  2 ) if there is  > 0 such that  () ≥ 0, ∀ ∈  1 with ‖‖ ≤   () ≤ 0, ∀ ∈  2 with ‖‖ ≤ .
Proof.From (H4) and the continuity of the potentials  and  we have that, for some  > 0, there exists a positive constant  3 such that Thus, by the Sobolev inequality [1] and (H1), for (, V) ∈ , we obtain as ‖(, V)‖ → ∞.Hence, we have that J is coercive in .
Hence we have the following direct sum: If (, V) ∈ , from Proposition 9 in [4], we get Moreover, if (, V) ∈ , by Proposition 9 in [4], we have Lemma 5. Assume that (H1)-(H3) hold.Then the functional J has a local linking at the origin with respect to  =  ⊕ .
Let  be a real Banach space and  ∈  1 (, R).Suppose  is an isolated critical point of  with () =  and  is a neighborhood of , containing the unique critical point; the group is called the th critical group of  at , where   = { ∈  : () ≤ } and   (⋅, ⋅) is the th singular relative homology group with integer coefficients.

The Proof of Theorem 1
We say that  is a homological nontrivial critical point of  if at least one of its critical groups is nontrivial.By [16], we have the following abstract critical point theorem.
Lemma 7 (see [16]).Let  be a real Banach space and let Φ ∈  1 (, R) satisfy the (.) condition and be bounded from below.If Φ has a critical point that is homologically nontrivial and is not the minimizer of Φ, then Φ has at least three critical points.
Proof of Theorem 1.By Lemmas 3 and 4, J is coercive and satisfies the (P.S) condition.Hence J is bounded below.By Lemma 8, (0, 0) ∈  is homologically nontrivial critical point of J but not a minimizer.Then the conclusion follows from Lemma 7.