Existence and multiplicity results for a coupled system of Kirchhoff type equations

This paper deals with a coupled system of Kirchhoff type equations in R3. Under suitable assumptions on the potential functions V(x) and W(x), we obtain the existence and multiplicity of nontrivial solutions when the parameter λ is sufficiently large. The method combines the Nehari manifold and the mountain-pass theorem.

In recent years, many papers have extensively considered the scalar Kirchhoff equation where Ω ⊂ R 3 is a smooth bounded domain, one can see [1,4,6,11,12,15] and the references therein.Problem (1.1) is related to the stationary analogue of the equation Corresponding author.Email: dengfeng1214@163.com(D.Lü), jhxmath@sina.cn(J.Xiao) 2 D. Lü and J. Xiao which was proposed by Kirchhoff in [8] as an extension of the classical d'Alembert wave equation for free vibrations of elastic strings.Kirchhoff's model considers the changes in length of the string produced by transverse vibrations.
There are also many works on the existence and multiplicity results for the scalar case of (K) where f is a subcritical function and satisfies certain conditions.We would mention the recent paper [14], by applying symmetric mountain-pass theorem, the author obtained the existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.3).Subsequently, Liu and He [9] proved the existence of infinitely many high energy solutions for (1.3) when f is a subcritical nonlinearity which does not need to satisfy the usual Ambrosetti-Rabinowitz conditions.Further related results can be seen in [7,10,13] and the references therein.The purpose of this paper is to study the existence and multiplicity results for a coupled system of Kirchhoff type equations in R 3 .To the best of our knowledge, problem (K) λ has not been considered before, the main difficulties lie in the appearance of the non-local term and the lack of compactness due to the unboundedness of the domain R 3 .Motivated by the work mentioned above, we will get the existence and multiplicity results of nontrivial solutions for λ large enough by exploiting the Nehari manifold method and the mountain-pass theorem.
Before stating our main results, we need to introduce some assumptions and notations: where L denotes the Lebesgue measure in R 3 .
The hypothesis (A 2 ) was first introduced by Bartsch and Wang [3] in the study of a nonlinear Schrödinger equation.Let )dx respectively.For any given λ > 0, we consider the Hilbert space E := E V × E W endowed with the norm The energy functional associated with (K) λ is defined on E by where Υ(w) = R 3 |∇w| 2 dx.In view of the assumptions (A 1 ) and (A 2 ), the energy functional I λ (u, v) is well defined and belongs to C 1 (E, R).It is well known that the weak solutions of problem (K) λ are the critical points of the energy functional I λ (u, v).
The main results we get are the following: Theorem 1.1.Suppose that (A 1 ) and (A 2 ) hold.Then there is λ * > 0 such that for all λ ≥ λ * , the system (K) λ has a ground state solution.
Theorem 1.2.Suppose that (A 1 ) and (A 2 ) hold.Then for any given k ∈ N, there exists Λ k > 0 such that for each λ ≥ Λ k , the system (K) λ possesses at least k pairs of nontrivial solutions.
This paper is organized as follows.In Section 2, we will prove some important lemmas that will be used for the proofs of the main results.Section 3 is devoted to the proofs of Theorems 1.1 and 1.2.

Some preliminary lemmas
In this paper, C, C 1 , C 2 , . . .denote positive (possibly different) constants.→ (respectively ) denotes strong (respectively weak) convergence.o n (1) denotes o n (1) → 0 as n → ∞.B r denotes a ball centered at the origin with radius r > 0. For a given set K ⊂ R 3 , we set K c = R 3 \K.We define the minimax c λ as where N λ denotes the Nehari manifold associated with I λ given by and •, • is the duality product between E and its dual space Hereafter, we suppose that (A 1 ) and (A 2 ) are satisfied.
Proof.First, by Young's inequality, we get then by the continuity of the Sobolev embedding where C > 0 is independent of λ.So, by (2.2), for any (u, v) ∈ N λ we have On the other hand, we have Hence, (u n , v n ) λ < σ 1 for n large, then by (2.4) which implies (u n , v n ) λ → 0 as n → ∞ and c = 0, it follows that (ii) holds for c 0 = σ 2 1 /2.
It is easy to see that for any (ϕ, ψ) ∈ E, we have Therefore (u, v) is a critical point of I λ .

6
D. Lü and J. Xiao By the Brézis-Lieb lemma, we have that To show (2.10) we observe Thus from (2.12) we obtain (2.10).

Proof of the main results
We begin with the following lemma.
The following lemma implies that I λ possesses the mountain-pass geometry.
Lemma 3.2.The functional I λ satisfies the following conditions.
Proof of Theorem 1.1.By Lemma 3.2, the functional I λ satisfies the mountain-pass geometry, then using a version of the mountain-pass theorem without (PS) condition, there exists a (PS)- Furthermore, by Lemma 2.5 we have that I λ (u, v) = 0.By Lemma 2.1, we know that (u, v) = (0, 0), then (u, v) ∈ N λ , and using Fatou's lemma we get Hence, I λ (u, v) ≤ c λ .On the other hand, from the definition of c λ , we have To prove Theorem 1.2 we need the following version of the symmetric mountain-pass theorem [2].Theorem 3.3.Let X be a real Banach space and W ⊂ X a finite dimensional subspace.Suppose that J ∈ C 1 (X, R) is an even functional satisfying J(0) = 0 and (a) there exists a constant ρ > 0 such that J| ∂B ρ (0) ≥ 0; (b) there exists M 0 > 0 such that sup z∈W J(z) < M 0 .If J satisfies (PS) c for any 0 < c < M 0 , then J possesses at least dim W pairs of nontrivial critical points.
In view of Lemma 3.1, there exists Λ k > 0 such that I λ satisfies (PS) c for any c ≤ M k and λ ≥ Λ k .Thus, for any fixed λ ≥ Λ k we may apply Theorem 3.3 to obtain k pairs of nontrivial solutions.Theorem 1.2 is proved.