Multiple Solutions For a Quasilinear Schrödinger System of Kirchhoff Type With Critical Exponents

Ω |∇v|dx ) ∆v − a[∆(v)]v = λFv(x, u, v) +η β α+βu 2α|v|2(β−1)v in Ω u = v = 0 on ∂Ω, (1.1) where Ω ⊂ R(N ≥ 3) is a bounded smooth domain, a, b > 0, α, β > 1 with α+β = 2∗ := 2N N−2 , ∇F = (Fu, Fv) is the gradient of a C 1 function F : Ω×R → R with respect to (u, v). When a(∆(u)) = a(∆(v)) = 0, system (1.1) reduces to standard nonlocal problem which is related to the stationary problem of a model presented by Kirchhoff [13]. Recently, Kirchhoff type problems have been studied in many papers, we refer to [5,6,7,18,9,22,23] in which different methods have been used to get the existence and multiplicity of solutions. On the other hand, problem (1.1) without nonlocal term arises naturally from finding the standing wave solutions for quasilinear Schrödinger equations of the form


Introduction and main results
In this article, we are concerned with the multiplicity of nontrivial solutions for the following nonlocal Schrödinger system where Ω ⊂ R N (N ≥ 3) is a bounded smooth domain, a, b > 0, α, β > 1 with α+ β = 2 * := 2N N −2 , ∇F = (F u , F v ) is the gradient of a C 1 function F : Ω× R 2 → R with respect to (u, v).
When a(∆(u 2 )) = a(∆(v 2 )) = 0, system (1.1) reduces to standard nonlocal problem which is related to the stationary problem of a model presented by Kirchhoff [13].Recently, Kirchhoff type problems have been studied in many papers, we refer to [5,6,7,18,9,22,23] in which different methods have been used to get the existence and multiplicity of solutions.
On the other hand, problem (1.1) without nonlocal term arises naturally from finding the standing wave solutions for quasilinear Schrödinger equations of the form where κ is a real constant, V is a given potential, g and h are real functions.
The study of This type of equations is motivated by its various applications, for example, the case h(s) = s was used to model the time evolution of the condensate wave function in superfluid film, and is called the superfluid film equation in fluid mechanics by Kurihura [14]; in the case h(s) = (1 + s) 1 2 , equation (1.2) was used as a model of the self-channeling of ahigh-power ultra short laser in matter, see [2,25] and the references therein.One of the main difficulties of the quasilinear problem with nonhomogeneous term [∆(u 2 )]u is that there is no suitable space on which the energy functional is well defined.There have been several approaches used in recent years to overcome the difficulties such as minimizations [19,24], the Nehari or Pohozaev manifold [20,26], and change of variables [1,8,21,27].The critical problems involving nonlocal operators create many difficulties in applying variational methods, these is due to the lack of compactness of the imbedding H 1 0 (Ω) ֒→ L 2 * (Ω) and the Palais-Smale condition fails.In a recent paper [12], D. S. Kang establish a variant of concentration compactness principle related to critical elliptic systems, which is based on the ideas by P. L. Lions [16,17].This result is very useful for the study of the existence of solutions for critical elliptic systems (see e.g., [11]).Motivated by the above, our purpose is to establish the existence of a sequence of solutions for system (1.1).We will assume that the function F satisfies the following conditions.
(F 1 ) lim Let H 1 0 (Ω) be the usual Sobolev space defined as the completion of Then X is a Hilbert space with respect to the inner product defined by and equipped with the norm

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The energy functional I λ,η : X → R corresponding to system (1.1) is given by Note that a major difficulty associated with (1.1) is that the functional I λ,η is not well defined in general, for instance, in X.To overcome this difficulty, we use an argument developed by Colin and Jeanjean [8].We make the changing of variables (u, v) = (f (w), f (z)), where f is given by Some properties of the function f are given in the following lemma.
Lemma 1.1.Concerning the function f (t) and its derivative satisfy the following properties: (f 1 ) f is uniquely defined, C ∞ and invertible; The function f 2 is strictly convex; (f 10 ) There exists a positive constant C > 0 such that So, by the change of variables, from I λ,η , we can define the following functional Then Φ λ,η is well defined.In view of assumptions, it standard to see that Φ λ,η ∈ C 1 (X, R) and its derivative at (ϕ, ψ) ∈ X is given by The main results of this paper are the following theorems.

Preliminary lemmas and proof of main results
We start with stating a few known results and giving a preliminary lemmas which we need in our argument.First we recall a variant of concentration compactness principle related to critical elliptic systems of D. S Kang [12].
ν in the sense of measures, where µ and ν are nonnegative bounded measures on R N .Then there exist an at most countable set J and families {x j } j∈J ⊂ R N and {µ j } j∈J , {ν j } j∈J ⊂ [0, +∞) such that where δ xj is the Dirac mass at x j and S α,β is given by and Then, by (2.1) and (2.2), for any ε > 0, we can find C 0 (ε) > 0 such that (2.3) Lemma 2.2.Assume that (F 0 ) − (F 1 ) hold.Then for any λ, η > 0, the functional On the other hand, we have Combine this with (2.2), for n large enough This last inequality shows that {(w n , z n )} is bounded in X. Therefore {w n } and {z n } are bounded in H 1 0 (Ω) and hence {(f 2 (w n ), f 2 (z n ))} is bounded in X.Then passing to a subsequence if necessary, we may assume that (2.7) By using the fact that f is continuous, it follows that (f in the sense of measures, where µ and ν are nonnegative bounded measures on R N .According to Lemma 2.1, there exist an at most countable set J and families For ε > 0 and j ∈ J denote By (2.5), w n φ j ε , z n φ j ε is bounded in X and therefore In view of Lemma 1.1 (f 5 ), Hölder's inequality and Lebesgue's dominated convergence theorem,

Multiple Solutions
195 Hence, up to subsequence lim and we also have (2.12) In the similar way, we get (2.15) Furthermore, by the continuity of F u and F v , we have Therefore, by (2.1), (2.15) and Egorov's theorem, we obtain and hence (2.17) Tacking account that it follows from (2.8)-(2.14)and (2.17) that By (2.9), we conclude that or ν j = 0. (2.18) Suppose by contradiction that ν j ≥ aS α,β 2η N 2 for some j ∈ J.Then, by (2.3) with -(2.9) and using the fact that 0 ≤ φ j ε ≤ 1, which is impossible.Therefore ν j = 0 and hence By the weak lower semicontinuity of the norm and f ∈ C ∞ , we entail that On the other hand, up to subsequence, Brezis-Lieb's Lemma [3] leads to Combining this with (2.21), we obtain where w := 1 + 2f 2 (w)f (w) and z := 1 + 2f 2 (z)f (z).Using the same arguments as above, we can prove that From this and (2.22), we deduce that (w n , z n ) → (w, z) strongly in X.This completes the proof of Lemma 2.2.✷ Now we use minimax procedure to prove Theorem 1.2.For a Banach space X, let Σ = {E ⊂ X\{0} : E is closed in X and symmetric with respect to the origin}.
For each E ∈ Σ, define If there is nomapping ϕ as above for any k ∈ N, then γ(E) = +∞.Set This next proposition is a version of the symmetric mountain-pass lemma [10].
Proposition 2.3.Let X be an infinite dimensional space and Φ ∈ C 1 (X, R) and assume the following assertions holds.
(i) Φ is even, Φ(0) = 0, bounded from below and satisfies the (P S c ) condition for c < c, for some c > 0; (ii) For each k ∈ N there exists Then, either (R 1 ) or (R 2 ) below holds.
0 and {v k } converges to a non-zero limit.