Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

Theorem 1.1

Suppose that (B), (V) and (f₀)-(f₄) hold. Then system (1.1) has a ground state solution z̃ such that Φ(z̃) = inf_ℳ Φ > 0.

Theorem 1.2

Suppose that (B), (V), (f₀)-(f₂) and (f₅) are satisfied. Then system (1.1) has a ground state solution z̃ such that Φ(z̃) = inf_𝒩 Φ > 0.

On multiplicity results of solutions we have the following theorems.

Theorem 1.3

Assume that (B), (V) and (f₀)-(f₄) hold, and if H(x, z) is even in z, then system (1.1) has infinitely many geometrically distinct solutions.

Theorem 1.4

Assume that (B), (V), (f₀)-(f₂) and (f₅) are satisfied, and if H(x, z) is even in z, then system (1.1) has infinitely many geometrically distinct solutions.

As observed in [33], the first author and co-author only proved the existence result of nontrivial solution by using generalized linking theorem under the conditions of Theorem 1.2, and the other related results are all unknown. Compared to the result in [33], the results obtained in this paper can be viewed as a continution of [33], and seem more delicate.

To prove Theorem 1.1 and Theorem 1.2, some arguments are in order. Here we first introduce the problem of ground state solution. For the Theorem 1.2, based on the result in [33], we can see that 𝒩 ≠ ∅. By using minimization method and concentration compactness argument, the conclusion for ground state solution in Theorem 1.2 holds. However, Theorem 1.1 seems more complicated than Theorem 1.2, and there are many new difficulties. Generally speaking, ℳ contains infinitely many elements of E, while 𝒩 may contain only one element. So it becomes more difficult to find a ground state solution z̃ which satisfies Φ(z̃) = inf_ℳ Φ than one that satisfies Φ(z̃) = inf_𝒩 Φ. Additionally, we note that the variational structure of system (1.1) is strongly indefinite, thus the usual Nehari manifold method cannot be applied directly. To overcome these difficulties, in the spirit of [20, 24] we choose the generalized Nehari manifold ℳ to work. More precisely, we intend to make use of the non-Nehari manifold method developed by Tang [25] to complete the proof of Theorem 1.1. The main idea of this method is to construct a minimizing Cerami sequence for energy functional Φ outside ℳ by using the diagonal method and linking argument.

We would like to point out that the global super-quadratic (SQ) is indispensable in verifying the linking geometry structure and constructing Cerami sequence by the diagonal method and linking argument, see [37, 40]. Unfortunately, in the present paper we have no global information on the nonlinearity like (SQ), then the non-Nehari manifold method seems not work to our problem under the local super-quaratic condition. So, some new methods and techniques need to be introduced. Motivated by [26], we present a perturbation approach by adding a perturbation term of power function. More precisely, for μ ∈ (0, 1] and p ∈ (2, 2^*), we consider the following perturbation problem

and its associated functional is as follows

In such a way, the modified nonlinearity satisfies the global super-quadratic condition (SQ), then, by using the non-Nehari method in [25] and concentration compactness principle, we can obtain a ground state solution z_μ of the perturbation problem. Finally, by passing to the limit and by some special techniques, we show the convergence as μ → 0 of {z_n} towards to a ground state solution of the original problem.

For the sake of completeness, next we consider the multiplicity results of system (1.1). In view of Theorem 1.1 and Theorem 1.2, we know that 𝒩 ≠ ∅. To prove the existence of infinitely many geometrically distinct solutions, we choose a subset ℱ of 𝒩 such that ℱ = −ℱ and each orbit 𝒪(w) ⊂ 𝒩 has a unique representative in ℱ due to Φ(z) = Φ(−z), and then show that the set ℱ is infinite. To do this, inspired by [24, 26], using some arguments about deformation type and Krasnoselskii genus, we find infinitely many geometrically distinct solutions.

The remainder of this paper is organized as follows. In Section 2, we introduce the variational setting of the problem and present some useful preliminaries. In Section 3, we prove that the perturbation problem has a ground state solution. In Section 4, we give the proof of the existence of ground state solutions in Theorem 1.1 and Theorem 1.2, respectively. At last, the existence of infinitely many geometrically distinct solutions is established in Section 5.

Lemma 2.1

Assume (B) and (V) are satisfied. Then operator A is a selfadjoint operator on L² := L²(ℝ^N, ℝ²) with domain 𝓓(A) := H²(ℝ^N, ℝ²).

Lemma 2.2

Assume (B) and (V) are satisfied. The following two conclusions hold:

1. σ(A) = σ_e(A), i.e., A has only essential spectrum;

2. σ(A) ⊂ ℝ ∖ (−a, a) and σ(A) is symmetric with respect to origin.

Observe that, it follows from Lemma 2.1 and 2.2 that the space L² possesses the following orthogonal decomposition

such that A is negative definite (resp. positive definite) in L⁻(resp. L⁺). Let ∣A∣ denote the absolute value of A and ∣A∣^1/2 be the square root of ∣A∣. Let E = 𝓓(∣A∣^1/2) be the Hilbert space with the inner product

and norm ∥z∥ = (z, z)^1/2. Moreover, it is obvious that E possesses the following decomposition

which is orthogonal with respect to the inner products (⋅, ⋅)₂ and (⋅, ⋅). Note that E = H¹ := H¹(ℝ^N, ℝ²) and ∥ ⋅ ∥ is equivalent to the usual norm of H¹ (see [32]). Then E embeds continuously into L^q for all q ∈ [2, 2^*] and compactly into Llocq for all q ∈ [1,2^*). Moreover, there exists a positive constant π_q > 0 such that for all z ∈ E

On the one hand, by virtue of (f₁) and (f₂), for any ϵ > 0, there exists a positive constant c_ϵ such that

On the other hand, it follows from (f₃) that

In fact, given z ≠ 0, (f₃) implies that

This, together with the monotonicity of h(x, ∣z∣), implies that

On E we define the energy functional Φ corresponding to system (1.1) as follows

By the above facts and some standard arguments, we can easily see that Φ ∈ C¹(E, ℝ) and the critical points of Φ are solutions of system (1.1) (see [6, 28]), and for z, φ ∈ E

Now we discuss the linking geometry structure of the energy functional Φ.

Lemma 2.3

Suppose that (f₁) and (f₂) are satisfied. Then there exists ρ > 0 such that κ := inf_SρΦ > 0, where S_ρ := {z ∈ E⁺, ∥z∥ = ρ}.

The proof of Lemma 2.3 is standard, and the details can be seen in [36] and hence is omitted.

Without loss of generality, we can assume that Ω ⊂ ℝ^N is a bounded domain. We choose ẽ ∈ C0∞ (ℝ^N) ∩ C0∞ (Ω) such that ∥ẽ⁺∥² − ∥ẽ⁻∥² = (Aẽ, ẽ)₂ ≥ 1, which implies that ẽ⁺ ≠ 0. Based on this fact, using the special technique as in [33], we can obtain the following lemma, which is very critical in our arguments, the proof can be found in [33].

Lemma 2.4

Suppose that (f₀)-(f₂) are satisfied. Then sup Φ(E⁻ ⊕ ℝ⁺ ẽ⁺) < ∞, and there is R_ẽ > 0 such that

Lemma 2.5

Suppose that (f₀)-(f₂) are satisfied. Then ℳ ≠ ∅.

Proof

Let E(ẽ⁺) = E⁻ ⊕ ℝ⁺ẽ⁺. It follows from Lemma 2.4 that there exists R_ẽ > 0 such that Φ(z) ≤ 0 for z ∈ E(ẽ⁺) ∖ B_Rẽ(0). Moreover, Lemma 2.3 implies that Φ(tẽ⁺) > 0 for small t > 0. Thus, 0 < supΦ(E(ẽ⁺)) < ∞. It is easy to see that Φ is weakly upper semi-continuous on E(ẽ⁺), therefore, Φ(z₀) = supΦ(E(ẽ⁺)) for some z₀ ∈ E(ẽ⁺). This z₀ is a critical point of Φ∣_E(ẽ⁺), so Φ′(z₀)z₀ = Φ′(z₀)w = 0 for all w ∈ E(ẽ⁺). Consequently, z₀ ∈ ℳ ∩ E(ẽ⁺), and ℳ ≠ ∅.□

We recall that a functional Φ ∈ C¹(E, ℝ) is said to be weakly sequentially lower semi-continuous if for any u_n ⇀ u in E one has Φ(u) ≤ lim infn→∞ Φ(u_n), and Φ′ is said to be weakly sequentially continuous if limn→∞ Φ′(u_n)φ = Φ′(u)φ for each φ ∈ E. We recall that a sequence {u_n} ⊂ E is called Cerami sequence for Φ at the level c ((C)_c-sequence in short) if

We say that Φ satisfy the (C)_c-condition if any (C)_c-sequence has a convergent subsequence.

To prove the main results, we need the following generalized linking theorem in [17].

Lemma 2.6

Let X be a real Hilbert space with X = X⁻ ⊕ X⁺, and let Φ ∈ C¹(X, ℝ) be of the form

Suppose that the following assumptions are satisfied:

1. Ψ ∈ C¹(X, ℝ) is bounded from below and weakly sequentially lower semi-continuous;

2. Ψ′ is weakly sequentially continuous;

3. there exist R > ρ > 0 and e ∈ X⁺ with ∥e∥ = 1 such that

   κ:=infΦ(Sρ+)>supΦ(∂Q),

   where

   Sρ+=u∈X+:∥u∥=ρ,Q=v+se:v∈X−,s≥0,∥v+se∥≤R.

Then there exist a constant c ∈ [κ, sup Φ(Q)] and a sequence {u_n} ⊂ X satisfying

Lemma 3.1

Ψ_μ is weakly sequentially lower semi-continuous. Ψ′_μ is weakly sequentially continuous.

Lemma 3.2

Suppose that (f₀)-(f₃) are satisfied. Let z ∈ E, w ∈ E⁻ and t ≥ 0, we have

In particular, let z ∈ ℳ_μ, w ∈ E⁻ and t ≥ 0. There holds

Proof

Observe that,

where

Since g_μ(x, s) is increasing in s on (0, +∞) due to (f₃), we can obtain 𝓖_μ(x, t) ≤ 0 for t ≥ 0 by using some arguments in [37, 40]. So, we get the first conclusion from the above formula. If z ∈ ℳ_μ, then Φ′_μ(z)z = Φ′_μ(z)w = 0, then the second conclusion holds.□

For convenience of notation, we write E(z) := E⁻ ⊕ ℝ⁺z = E⁻ ⊕ ℝ⁺z⁺ for z ∈ E ∖ E⁻. Let z ∈ ℳ_μ, then Lemma 3.2 implies that z is the global maximum of Φ_μ∣_E(z). Next we shall verify that Φ_μ possesses the linking structure.

Lemma 3.3

Suppose that (f₀)-(f₃) are satisfied. Then there exist positive constants ρ and α both independent of μ ∈ (0, 1] such that

1. there holds: m_μ = inf_ℳμ Φ_μ ≥ κ := inf_SρΦ_μ ≥ α, where S_ρ := {z ∈ E⁺, ∥z∥ = ρ}.

2. ∥z⁺∥ ≥ max{∥z−∥,2mμ} for all z ∈ ℳ_μ.

Proof

1. For z ∈ E⁺ and μ ∈ (0, 1], by (2.1) and (2.2), we obtain

   Φμ(z)=12∥z∥2−∫RNH(x,z)−μp∫RN|z|p≥12∥z∥2−ϵ|z|22−cϵ+1p|z|pp≥12−ϵπ2−2∥z∥2−πp−pCϵ∥z∥p.

   It is easy to see that there exist positive constants ρ and α both independent of μ such that inf_SρΦ ≥ α due to the arbitrariness of ϵ > 0. So the second inequality holds. Note that for every z ∈ ℳ_μ there exists s > 0 such that sz⁺ ∈ E(z) ∩ S_ρ. Hence, by Lemma 3.2 we know that the first inequality holds.

2. It follows from (2.3) and μ ∈ (0, 1] that Ψ_μ(z) ≥ 0. For z ∈ ℳ_μ, then we have Φ_μ(z) ≥ m_μ and

   mμ≤12(∥z+∥2−∥z−∥2)−Ψμ(z)≤12(∥z+∥2−∥z−∥2),

   and this implies that ∥z⁺∥ ≥ max{∥z−∥,2mμ}. □

Lemma 3.4

Suppose that (f₀)-(f₃) are satisfied. Then for any e ∈ E⁺, sup Φ_μ(E⁻ ⊕ ℝ⁺ e) < ∞, and there is R_e > 0 such that

In particular, there is a R₀ > ρ such that sup Φ_μ(∂ Q_R) ≤ 0 for R ≥ R₀, where

Proof

Since the modified nonlinearity G_μ(x, z) satisfies the global super-quadratic condition (SQ), the proof is standard, see [36, 40]. So we omit it here.□

Employing Lemmas 2.6, 3.1, 3.3 and 3.4, we have

Lemma 3.5

Suppose that (f₀)-(f₃) are satisfied. Then there exist a constant ĉ_μ ∈ [κ, sup Φ_μ(Q_R)] and a (C)_ĉμ-sequence {z_n} ⊂ E satisfying

To prove the existence of ground state solutions for the perturbation problem (1.3), next we construct a special Cerami sequence by using diagonal method (see [25]), which is very important in our arguments.

Lemma 3.6

Suppose that (f₀)-(f₃) are satisfied. Then there exist a constant c̃_μ ∈ [κ, m_μ] and a (C)_c̃μ-sequence {z_n} ⊂ E such that

Proof

Choose ξ_k ∈ ℳ_μ such that

By virtue of Lemma 3.3, ∥ξk+∥≥2mμ>0. Set ek=ξk+/∥ξk+∥. Then e_k ∈ E⁺ and ∥e_k∥ = 1. From Lemma 3.4, it follows that there exists R_k such that supΦ_μ(∂ Q_k) ≤ 0, where

Hence, using Lemma 3.5 to the above set Q_k, there exist a constant c_μ,k ∈ [κ, sup Φ_μ(Q_k)] and a sequence {z_k,n}_n∈ℕ ⊂ E satisfying

On the other hand, by Lemma 3.2, one can get that

Since ξ_k ∈ Q_k, it follows from (3.4)) and (3.6) that Φ_μ(ξ_k) = sup Φ_μ(Q_k). Hence, by (3.3) and (3.5), one has

We can choose a sequence {n_k} ⊂ ℕ such that

Let z_k = z_k,nk, k ∈ ℕ. Then, up to a subsequence, we have

□

Similar to the proof of Lemma 2.5, we have the following result.

Lemma 3.7

Suppose that (f₀)-(f₃) are satisfied. Then for any z ∈ E ∖ E⁻, ℳ_μ ∩ E(z) ≠ ∅, i.e., there exist t_μ > 0 and w_μ ∈ E⁻ such that t_μz + w_μ ∈ ℳ_μ.

Lemma 3.8

Suppose that (f₀)-(f₃) are satisfied. If {z_n} ⊂ E satisfies (1 + ∥z_n∥)Φ′_μ(z_n) → 0 and Φ_μ(z_n) is bounded from above, then {z_n} is bounded. In particular, any (C)_c-sequence of Φ_μ at level c ≥ 0 is bounded.

Proof

Let {z_n} ⊂ E be such that

for some C > 0. Suppose to the contrary that ∥z_n∥ → ∞ as n → ∞. Setting w_n = z_n/∥z_n∥, then ∥w_n∥ = 1. After passing to a subsequence, we assume that w_n ⇀ w in E, w_n → w in Llocq for 2 ≤ q < 2^*, and w_n(x) → w(x) a.e. on ℝ^N. Let

If δ = 0, by Lions’ concentration compactness principle (see [14, 28]), then wn+ → 0 in L^q for any 2 < q < 2^*. It follows from (2.2) that for any s > 0,

Observe that, from (3.1) and (3.7), we deduce that

Let t_n = s/∥z_n∥, then by (3.8) we have

On the other hand, by (2.3) we get 〈Φμ′(zn),zn〉≤∥zn+∥2−∥zn−∥2, which implies that

This shows that ∥wn+∥2≥c1 for some c₁ > 0. Hence, (3.9) yields a contradiction if s is large enough. Then δ > 0. Up to a subsequence, we assume k_n ∈ ℤ^N such that

Let w̃_n(x) = w_n(x + k_n). By the periodicity of b⃗ and V, we have that ∥w_n∥ = ∥w̃_n∥ = 1 and

Therefore, passing to a subsequence, w~n+ → w̃⁺ in Lloc2 and w̃⁺ ≠ 0. Note that if w̃ ≠ 0, then ∣z_n(x + k_n)∣ = ∣w̃_n(x)∣∥z_n∥ → ∞. Since G_μ satisfies condition (SQ), then it follows from Fatou’s lemma that

which is a contradiction. So, {z_n} is bounded in E.□

Next we show the existence of ground state solutions of the perturbation problem (1.3).

Lemma 3.9

The perturbation problem (1.3) possesses a ground state solution, and m_μ is attained for all μ ∈ (0, 1].

Proof

Applying Lemma 3.6, we deduce that there exists a (C)_c̃μ-sequence {z_n} of Φ_μ such that

Lemma 3.8 shows that {z_n} is bounded. Let

By Lions’ concentration compactness principle (see [14, 28]), we have z_n → 0 in L^q for 2 < q < 2^*. Moreover, by (2.2) we get

and consequently

which is a contradiction. Thus δ > 0. Up to a subsequence, we assume k_n ∈ ℤ^N such that

Let us define z̃_n(x) = z_n(x + k_n) so that

Since the periodicity of b⃗ and V, we have ∥z̃_n∥ = ∥z_n∥ and

Passing to a subsequence, we assume that z̃_n ⇀ z̃ in E, z̃_n → z̃ in Llocq for 2 ≤ q < 2^*, and z̃_n(x) → z̃(x) a.e. on ℝ^N. Hence it follows from (3.10) and (3.11) that z̃ ≠ 0 and Φ′_μ(z̃) = 0. This shows that z̃ ∈ ℳ_μ and Φ_μ(z̃) ≥ m_μ. On the other hand, by (2.3), (3.11) and Fatou’s lemma, we get

This shows that Φ_μ(z̃) ≤ m_μ. Hence Φ_μ(z̃) = m_μ = inf_z∈ℳμ Φ_μ, and z̃ is a ground state solution of the perturbation problem (1.3).□

Proof

Proof of Theorem 1.1. We note that ℳ ≠ ∅ from Lemma 2.5. Let z₀ ∈ ℳ, then Φ(z₀) := c^* ≥ 0. In view of Lemma 3.7, there exist t_μ > 0 and w_μ ∈ E⁻ such that t_μz₀ + w_μ ∈ ℳ_μ. On the other hand, (3.2) also holds for μ = 0 by (f₃). Thus, from Lemma 3.2 and Lemma 3.3, it follows that, for μ ∈ (0, 1],

Let {μ_n} ⊂ (0, 1] be a sequence such that μ_n → 0⁺ as n → ∞, and

We denote z_n := z_μn. In view of (4.2), we obtain

In what follows, we show that {z_n} is bounded in E. Suppose to the contrary that ∥z_n∥ → ∞ as n → ∞. Setting w_n = z_n/∥z_n∥, then ∥w_n∥ = 1. After passing to a subsequence, we assume that w_n ⇀ w in E, w_n → w in Llocq for 2 ≤ q < 2^*, and w_n(x) → w(x) a.e. on ℝ^N. Let

Similarly to the proof of Lemma 3.8, we can show that δ > 0.

Passing if necessary to a subsequence, we assume k_n ∈ ℤ^N such that

Let w̃_n(x) = w_n(x + k_n), since the periodicity of b⃗ and V, then ∥w_n∥ = ∥w̃_n∥ = 1 and

Up to a subsequence, we have w̃_n ⇀ w̃ in E, w̃_n → w̃ in Llocq for 2 ≤ q < 2^* and w̃_n → w̃ a.e on ℝ^N. Moreover, (4.4) shows that w̃ ≠ 0. Let z̃_n = z_n(x + k_n), then z̃_n/∥z_n∥ = w̃_n and w̃_n → w̃ a.e. on ℝ^N. For any φ ∈ C0∞ (ℝ^N), let φ_n = φ(x − k_n). By virtue of (4.2), we get

which implies that