Home Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
Article Open Access

Existence and multiplicity results for non-autonomous second-order Hamiltonian systems

  • Chungen Liu EMAIL logo and Yuyou Zhong
Published/Copyright: December 31, 2024
Download Article (PDF)
Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

In this article, using the least action principle and minimax methods in critical point theory, some existence and multiplicity results for periodic solutions of second-order Hamiltonian systems are obtained.

Keywords: periodic solutions; the least action principle; minimax methods; second-order Hamiltonian systems
MSC 2010: 34C25; 37J45; 34C52

1 Introduction and main results

Consider the following second-order Hamiltonian systems:

(1.1) u ¨ ( t ) = ∇ F ( t , u ( t ) ) , a.e. t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ˙ ( 0 ) − u ˙ ( T ) = 0 ,

where constant T > 0 and the function F : [ 0 , T ] × R N → R satisfies the following condition:

( A ) F ( t , x ) is measurable in t for each x ∈ R N , continuously differentiable in x for a.e. t ∈ [ 0 , T ] , and there exist a ∈ C ( R + ; R + ) and b ∈ L 1 ( [ 0 , T ] ; R + ) such that

∣ F ( t , x ) ∣ ≤ a ( ∣ x ∣ ) b ( t ) , ∣ ∇ F ( t , x ) ∣ ≤ a ( ∣ x ∣ ) b ( t ) , ∀ x ∈ R N and a.e. t ∈ [ 0 , T ] ,

where ∇ F ( t , x ) denotes the gradient of F ( t , x ) in x .

In the sequel of this article, we may as well suppose that F ( t , θ ) = 0 and define the usual Sobolev space as

H T 1 = { u : [ 0 , T ] → R N ∣ u is absolutely continuous , u ( 0 ) = u ( T ) , u ˙ ∈ L 2 ( [ 0 , T ] ; R N ) } ,

which is a Hilbert space with the norm

‖ u ‖ = ∫ 0 T ( ∣ u ˙ ( t ) ∣ 2 + ∣ u ( t ) ∣ 2 ) d t 1 2 , ∀ u ∈ H T 1 .

We denote by ⟨ ⋅ , ⋅ ⟩ and ( ⋅ , ⋅ ) the inner product in H T 1 and R N , respectively.

From [9], we can see that the corresponding functional φ of problem (1.1) defined in H T 1 is

(1.2) φ ( u ) = 1 2 ∫ 0 T ∣ u ˙ ( t ) ∣ 2 d t + ∫ 0 T F ( t , u ( t ) ) d t ,

which is continuously differentiable and weakly lower semi-continuous (w.l.s.c.) under the condition ( A ) . Furthermore, there holds

(1.3) ⟨ φ ′ ( u ) , v ⟩ = ∫ 0 T ( ( u ˙ ( t ) , v ˙ ( t ) ) + ( ∇ F ( t , u ( t ) ) , v ( t ) ) ) d t , ∀ u , v ∈ H T 1 .

So a solution of problem (1.1) corresponds to a critical point of φ .

Reviewing the history, the existence of periodic solutions for problem (1.1) was considered by using the least action principle or the minimax methods under various conditions on the potential function F . For examples, in [2], it was considered under the coercive-type condition. The Ambrossiti-Rabinowitz superquadratic condition was proposed on F in [13]. With the convex condition supposed on F , it was studied in [8]. It was considered under the bi-even subquadratic condition in [5], the bounded nonlinearity condition in [9], the γ -quasisubadditive potential condition in [17], the sublinear nonlinearity potential condition in [19], the linear nonlinearity potential condition in [12,20,25,26], the generalized sublinear nonlinearity potential condition in [23,24], the generalized linear nonlinearity potential condition in [27], and some other conditions in [1,4,7,16] and references therein.

When gradient ∇ F ( t , x ) is provided with bounded nonlinearity, i.e., there exists a function g ∈ L 1 ( [ 0 , T ] ; R + ) such that

(1.4) ∣ ∇ F ( t , x ) ∣ ≤ g ( t ) ,

for each x ∈ R N and a.e. t ∈ [ 0 , T ] , and F ( t , x ) satisfies the following condition:

(1.5) ∫ 0 T F ( t , x ) d t → ± ∞ , as ∣ x ∣ → + ∞ ,

Mawhin and Willem [9] proved that problem (1.1) possesses one periodic solution.

When gradient ∇ F ( t , x ) is sublinear nonlinearity, i.e., there exist f , g ∈ L 1 ( [ 0 , T ] ; R + ) and α ∈ [ 0 , 1 ) such that

(1.6) ∣ ∇ F ( t , x ) ∣ ≤ f ( t ) ∣ x ∣ α + g ( t )

for each x ∈ R N and a.e. t ∈ [ 0 , T ] , and F ( t , x ) satisfies the following condition:

(1.7) 1 ∣ x ∣ 2 α ∫ 0 T F ( t , x ) d t → ± ∞ , as ∣ x ∣ → + ∞ ,

Tang [19] obtained the existence of periodic solutions for problem (1.1), which is a generalization of Mawhin-Willem’s results in [9].

When α = 1 , condition (1.6) is called the linear-type condition, i.e., there exist f , g ∈ L 1 ( [ 0 , T ] ; R + ) such that

(1.8) ∣ ∇ F ( t , x ) ∣ ≤ f ( t ) ∣ x ∣ + g ( t ) , ∀ x ∈ R N , a.e. t ∈ [ 0 , T ] .

In this case, Zhao and Wu [25,26] proved that problem (1.1) has one periodic solution under the conditions

(1.9) ∫ 0 T f ( t ) d t < 12 T

and

(1.10) 1 ∣ x ∣ 2 ∫ 0 T F ( t , x ) d t → ± ∞ , as ∣ x ∣ → + ∞ .

When potential F ( t , x ) has the decomposition F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) , where F 1 ( t , x ) is ( λ , μ ) -subconvex in x , i.e., the function F 1 ( t , ⋅ ) : R N → R satisfies the following condition:

(1.11) F 1 ( t , λ ( x + y ) ) ≤ μ ( F 1 ( t , x ) + F 1 ( t , y ) ) ,

for some λ , μ > 0 and each x , y ∈ R N and gradient ∇ F 2 ( t , x ) is sublinear-type condition, i.e., F 2 ( t , x ) satisfies the condition (1.6), Wu and Tang [22] obtained the existence of periodic solutions for problem (1.1) under the following assumption:

(1.12) 1 ∣ x ∣ 2 α 1 μ ∫ 0 T F 1 ( t , λ x ) d t + ∫ 0 T F 2 ( t , x ) d t → + ∞ , as ∣ x ∣ → + ∞ .

When α = 1 , Zhao and Wu [25] considered the existence of periodic solutions for problem (1.1) where F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) , F 1 ( t , x ) is ( λ , μ ) -subconvex in x , ∇ F 2 ( t , x ) satisfies condition (1.6) with ∫ 0 T f ( t ) d t < 12 ⁄ T , and F ( t , x ) satisfies (1.12).

Wang and Zhang [23] obtained some results by using a control function h ( ∣ x ∣ ) instead of ∣ x ∣ in (1.6) and (1.7), where h satisfies the following conditions with α ∈ [ 0 , 1 ] .

( H α ) There exist constants C 0 > 0 , K 1 > 0 , K 2 > 0 , and a nonnegative function h ∈ C ( [ 0 , + ∞ ) ; [ 0 , + ∞ ) ) with the properties:

( i ) h ( s ) ≤ h ( t ) , ∀ s ≤ t , s , t ∈ [ 0 , + ∞ ) , ( ii ) h ( s + t ) ≤ C 0 ( h ( s ) + h ( t ) ) , ∀ s , t ∈ [ 0 , + ∞ ) , ( iii ) 0 ≤ h ( t ) ≤ K 1 t α + K 2 , ∀ t ∈ [ 0 , + ∞ ) , ( iv ) h ( t ) → + ∞ , as t → + ∞ .

Similar to the discussion of Meng-Tang’s results in [12,20], Zhang and Tang [27] also obtained some new results when using a control function h ( ∣ x ∣ ) instead of ∣ x ∣ , where h ( ∣ x ∣ ) satisfies ( H α ) with α ∈ [ 0 , 1 ] .

In this article, motivated by previous studies [4,10–12,20,23,25,27], we consider the existence and multiplicity of problem (1.1) with mixed nonlinearity. Precisely, we write the potential function in the form F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with the function F 1 ( t , x ) having the following conditions:

( H 1 ) F 1 ( t , ⋅ ) is ( λ , μ ) -subconvex with λ > 1 2 and μ < 2 λ 2 for a.e. t ∈ [ 0 , T ] .

( H 1 * ) F 1 ( t , ⋅ ) is ( λ , μ ) -subconvex with 0 < λ < 1 2 and 2 λ 2 < μ ≤ 1 2 for a.e. t ∈ [ 0 , T ] .

( H 2 ) There exist γ ∈ L 1 ( [ 0 , T ] ; R ) and ξ ∈ L 1 ( [ 0 , T ] ; R N ) where ∫ 0 T ξ ( t ) d t = 0 such that

(1.13) F 1 ( t , x ) ≥ ( ξ ( t ) , x ) + γ ( t ) , ∀ x ∈ R N and a.e. t ∈ [ 0 , T ] .

( H 2 * ) There exist γ ∈ L 1 ( [ 0 , T ] ; R ) and ξ ∈ L 1 ( [ 0 , T ] ; R N ) satisfying (1.13).

Assume that F 2 ( t , x ) satisfies the following conditions.

( H 3 ) There exist f , g ∈ L 1 ( [ 0 , T ] ; R + ) and h satisfies ( H α ) such that

(1.14) ∣ ∇ F 2 ( t , x ) ∣ ≤ f ( t ) h ( ∣ x ∣ ) + g ( t ) , ∀ x ∈ R N and a.e. t ∈ [ 0 , T ] .

( f 1 ) ‖ f ‖ ∞ < 2 π 2 ⁄ ( C 0 K 1 T 2 ) .

( H 3 * ) (1.14) holds with h satisfying (i), (ii), (iv) of ( H α ) and instead condition (iii) with the following condition:

( iii ′ ) There exist K 1 ′ , K 2 ′ ≥ 0 , K 1 , K 2 ≥ 0 , and α ∈ ( 1 ⁄ 2 , 1 ] satisfying 0 ≤ K 1 ′ t α + K 2 ′ ≤ h ( t ) ≤ K 1 t + K 2 , ∀ t ∈ [ 0 , + ∞ ) .

( H 4 ) There holds liminf ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t > C 0 2 T 2 ‖ f ‖ L 2 2 8 π 2 .

( H 4 * ) There holds liminf ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t > C 0 2 T 2 ‖ f ‖ L 2 2 8 π 2 − 4 C 0 K 1 T 2 ‖ f ‖ ∞ .

Together on the functions F 1 ( t , x ) and F 2 ( t , x ) , we have the following conditions.

( H 5 ) There holds liminf ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) 1 μ ∫ 0 T F 1 ( t , λ x ) d t + ∫ 0 T F 2 ( t , x ) d t > C 0 2 T 2 ‖ f ‖ L 2 2 8 π 2 .

( H 5 * ) There holds liminf ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) 1 μ ∫ 0 T F 1 ( t , λ x ) d t + ∫ 0 T F 2 ( t , x ) d t > C 0 2 T 2 ‖ f ‖ L 2 2 8 π 2 − 4 C 0 K 1 T 2 ‖ f ‖ ∞ .

( H 6 ) There holds liminf ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t − ∫ 0 T F 1 ( t , − x ) d t > C 0 2 T 2 ‖ f ‖ L 2 2 8 π 2 .

( H 6 * ) There holds liminf ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t − ∫ 0 T F 1 ( t , − x ) d t > C 0 2 T 2 ‖ f ‖ L 2 2 8 π 2 − 4 C 0 K 1 T 2 ‖ f ‖ ∞ .

( H 7 ) There exist positive integer k and r > 0 such that

− 1 2 ( k + 1 ) 2 ω 2 ∣ x ∣ 2 ≤ F ( t , x ) ≤ − 1 2 k 2 ω 2 ∣ x ∣ 2 , ∀ ∣ x ∣ ≤ r and a.e. t ∈ [ 0 , T ] ,

where ω = 2 π T .

Theorem 1.1

Assume that F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with the condition ( H 1 ) . We have two cases

Case 1: F i ( t , x ) , i = 1 , 2 satisfy ( H 3 ) with α ∈ [ 0 , 1 ) and ( H 5 ) .

Case 2: F i ( t , x ) , i = 1 , 2 satisfy ( H 3 ) with α = 1 , ( f 1 ) and ( H 5 * ) .

Then, problem (1.1) possesses a periodic solution that minimizes the function φ in H T 1 .

Remark 1

Theorem 1.1 generalizes Theorem 1.1 or Theorem 1.3 of [27]. When F 1 ( t , x ) = 0 , our Theorem 1.1 is Theorem 1.1 or is a special case of Theorem 1.3 of [27]. It cannot be covered by the results of [4], for example, take F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with

F 1 ( t , x ) = 4 + sin ∣ x ∣ 6 and F 2 ( t , x ) = 1 2 sin 2 π t T ln 3 2 ( 1 + ∣ x ∣ 2 ) + T 6 ln ( 1 + ∣ x ∣ 2 ) − 4 .

Then, we obtain ∣ ∇ F 2 ( t , x ) ∣ ≤ 3 4 ∣ sin 2 π t T ∣ ln 1 2 ( 1 + ∣ x ∣ 2 ) + T 6 for a.e. t ∈ [ 0 , T ] and all x ∈ R N . Therefore, we see that ( H 1 ) and ( H 3 ) hold with F 1 ( t , x ) is ( 1 , 1 ) -subconvex, f ( t ) = 3 4 ∣ sin 2 π t T ∣ , g ( t ) = T 6 , and h ( ∣ x ∣ ) = ln ( 1 + ∣ x ∣ 2 ) . By taking C 0 = 2 , we see that the condition ( H 5 ) is satisfied for any T ∈ ( 0 , 32 π 2 27 ) . But it does not satisfy the condition ( H 3 ) of Theorem 1.1 in [4] in general. We also note that F ( t , x ) does not satisfy ( H 5 ) of Theorem 1.2 of [4]. It is easy see that the function F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with

F 1 ( t , x ) = 4 + sin ∣ x ∣ 6 and F 2 ( t , x ) = 1 2 ln 2 ( 1 + ∣ x ∣ 2 ) + sin 2 π t T ln ( 1 + ∣ x ∣ 2 ) − 4

satisfies ( H 1 ) , ( H 3 * ) , ( H 5 * ) of Theorem 1.1 but it does not satisfy ( H 5 ) of Theorem 1.2 of [4].

Theorem 1.2

Assume that F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with the condition ( H 1 * ) . We have two cases

Case 1: F i ( t , x ) , i = 1 , 2 satisfy ( H 3 ) with α ∈ [ 0 , 1 ) and ( H 6 ) .

Case 2: F i ( t , x ) , i = 1 , 2 satisfy ( H 3 ) with α = 1 , ( f 1 ) and ( H 6 * ) .

Then, problem (1.1) possesses one periodic solution that minimizes the function φ in H T 1 .

Remark 2

Theorem 1.2 generalizes Theorem 1 of [10] or Theorem 1 of [11]. It cannot be covered by the results of [4]. For instance, the function F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with

F 1 ( t , x ) = − 4 + ∣ x ∣ 1 2 + sin ∣ x ∣ 4 and F 2 ( t , x ) = ∣ x ∣ 2 4 ln ( 100 + ∣ x ∣ 2 ) + sin 2 π t T ln ( 1 + ∣ x ∣ 2 ) + 4

satisfies all conditions of Theorem 1.2, but it does not satisfy the conditions of Theorem 1.1 and 1.2 in [4].

Theorem 1.3

Assume that F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) . We have three cases

Case 1: assume ( H 2 ) , ( H 3 ) with α ∈ [ 0 , 1 ) and ( H 4 ) .

Case 2: assume ( H 2 ) , ( H 3 ) with α = 1 , ( f 1 ) and ( H 4 * ) .

Case 3: assume ( H 2 * ) , ( H 3 * ) , ( f 1 ), and ( H 4 * ) .

Then, in every case of the aforementioned three cases, problem (1.1) possesses one periodic solution that minimizes the function φ in H T 1 .

Remark 3

Theorem 1.3 generalizes the results of Theorem 3 of [22] and Theorem 5 of [25]. It cannot be covered by the results of [4], for example, the function F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with

F 1 ( t , x ) = − 9 + ∣ x 1 ∣ + sin ∣ x ∣ 4 and F 2 ( t , x ) = 1 2 ln 2 ( 1 + ∣ x ∣ 2 ) + sin 2 π t T ln ( 1 + ∣ x ∣ 2 ) + 9 , x ∈ R N

satisfies all conditions of Theorem 1.3, but it does not satisfy the conditions of Theorems 1.1 and 1.2 in [4]. Our results (Theorems 1.1–1.3) also cannot be covered by the results of [6,14,21,28,29], which considered the ordinary p ( t ) -Laplacian differential system.

We now state a result about the multiplicity for problem (1.1). For this topic, we refer the articles [4,16,18,19,23,24,26] and references therein.

Theorem 1.4

Assume that F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) satisfies ( H 7 ) and the conditions of Theorem 1.3. Then, problem (1.1) possesses at least three distinct solutions in H T 1 .

Remark 4

There are many functions satisfying all the conditions of Theorem 1.4. For example, take the function F ( t , x ) = F 1 ( t , x ) + F 2 ( t , x ) with

F 1 ( t , x ) = 5 + ∣ x 1 ∣ + sin ∣ x ∣ 4 − sin 1 , x = ( x 1 , x 2 , … , x N ) ∈ R N , ∣ x ∣ > 1 , 5 − 1 4 ω 2 ∣ x ∣ 2 + 1 2 ω 2 + 2 + 2 5 ∣ x 1 ∣ ∣ x ∣ 4 − 1 4 ω 2 + 1 + 1 5 ∣ x 1 ∣ ∣ x ∣ 6 , ∣ x ∣ ≤ 1 , F 2 ( t , x ) = x 2 ln ( 100 + x 2 ) − 1 ln 101 , ∣ x ∣ > 1 , − 1 4 ω 2 ∣ x ∣ 2 + 1 2 ω 2 ∣ x ∣ 4 − 1 4 ω 2 ∣ x ∣ 6 , ∣ x ∣ ≤ 1 ,

and 0 < T < π 2 34 . It is easy to see that F ( t , x ) satisfies the conditions ( H 2 ) , ( H 3 ) , ( H 4 * ) , and ( H 7 ) with ξ ( t ) = ( sin 2 π T t , 0 , … , 0 ) , γ ( t ) = − 10 , f ( t ) = 4 , g ( t ) = 5 , h ( ∣ x ∣ ) = ∣ x ∣ ln ( 100 + ∣ x ∣ 2 ) , C 0 = K 1 = 1 , and k = 1 , r > 0 small enough. The condition ( f 1 ) also hold by the aforementioned assumption.

2 Preliminaries and proofs of theorems

In this section, we first introduce three lemmas that contribute to the proof of Theorems 1.1–1.4 as follows.

Lemma 2.1

[9] If u ∈ H ˜ T 1 , where H ˜ T 1 = { u ∈ H T 1 ∣ ∫ 0 T u ( t ) d t = 0 } , then Wirtinger’s inequality says

∫ 0 T ∣ u ( t ) ∣ 2 d t ≤ T 2 4 π 2 ∫ 0 T ∣ u ˙ ( t ) ∣ 2 d t ,

and the Sobolev inequality is

‖ u ‖ ∞ 2 ≤ T 12 ∫ 0 T ∣ u ˙ ( t ) ∣ 2 d t .

Lemma 2.2

[15] Assume V is a reflexive Banach space with norm ‖ ⋅ ‖ V , and let M ⊂ V be a weakly closed subset of V. Assume E : M → R ∪ + ∞ is coercive and (sequentially) w.l.s.c on M with respect to V, i.e., assume the following conditions are satisfied.

  1. E ( u ) → + ∞ as ‖ u ‖ V → + ∞ , u ∈ M .

  2. For all u ∈ M , all sequence ( u m ) in M such that u m ⇁ u weakly in V, there holds:

    E ( u ) ≤ liminf m → + ∞ E ( u m ) .

    Then, E is bounded from below on M and attains its infimum in M.

Lemma 2.3

[3] Let X be a Banach space with a direct sum decomposition X = X 1 ⊕ X 2 with k = dim X 2 < ∞ and φ ∈ C 1 ( X ; R ) with φ ( θ ) = 0 , which satisfies (PS) condition. Suppose that for some r > 0 ,

(2.1) φ ( u ) ≥ 0 , ∀ u ∈ X 1 w i t h ‖ u ‖ ≤ r , φ ( u ) ≤ 0 , ∀ u ∈ X 2 w i t h ‖ u ‖ ≤ r .

Suppose also that φ is bounded below and inf X φ < 0 . Then, φ possesses at least three distinct critical points.

Next, we give the proofs of our theorems. In the sequel, we denote by C a suitable positive constant, which may take different values in various estimations. Taking H T 1 = M = V and φ = E , we will use Lemma 2.2 to prove Theorems 1.1–1.3. Since φ is w.l.s.c. and continuously differentiable, we only need to claim that φ satisfies the condition of ( 1 ∘ ) of Lemma 2.2.

Proof of Theorem 1.1

As Theorem 1.1 stated, we divided our proof into two cases.

Proof of Case 1. Let β = log 2 λ ( 2 μ ) , by ( H 1 ) , we have 0 ≤ β < 2 . There exists a positive integer n so that for ∣ x ∣ > 1 ,

n − 1 < log 2 λ ∣ x ∣ ≤ n ,

which implies ∣ x ∣ β > ( 2 λ ) ( n − 1 ) β = ( 2 μ ) n − 1 and ∣ x ∣ ≤ ( 2 λ ) n . Therefore, combining conditions ( A ) and ( H 1 ) , we obtain

F 1 ( t , x ) ≤ 2 μ F 1 t , x 2 λ ≤ ⋯ ≤ ( 2 μ ) n F 1 t , x ( 2 λ ) n ≤ 2 μ ∣ x ∣ β a 0 b ( t )

for a.e. t ∈ [ 0 , T ] and every ∣ x ∣ > 1 , where a 0 = max 0 ≤ s ≤ 1 a ( s ) . Furthermore, we obtain

(2.2) F 1 ( t , x ) ≤ ( 2 μ ∣ x ∣ β + 1 ) a 0 b ( t )

for a.e. t ∈ [ 0 , T ] and every x ∈ R N .

Since F 1 ( t , ⋅ ) is ( λ , μ ) -subconvex, we obtain

F 1 ( t , λ u ¯ ) ≤ μ ( F 1 ( t , u ( t ) ) + F 1 ( t , − u ˜ ( t ) ) ) ,

with u = u ¯ + u ˜ where u ¯ = 1 T ∫ 0 T u ( t ) d t and u ˜ ∈ H ˜ T 1 , i.e.,

(2.3) F 1 ( t , u ( t ) ) ≥ 1 μ F 1 ( t , λ u ¯ ) − F 1 ( t , − u ˜ ( t ) ) .

It follows from the Sobolev inequality in Lemma 2.1, (2.2), and (2.3) that

(2.4) ∫ 0 T F 1 ( t , u ( t ) ) d t ≥ 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) d t − ∫ 0 T F 1 ( t , − u ˜ ( t ) ) d t ≥ 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) d t − 2 μ a 0 ∫ 0 T ∣ u ˜ ( t ) ∣ β b ( t ) d t − a 0 ∫ 0 T b ( t ) d t ≥ 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) − 2 μ a 0 ‖ u ˜ ‖ ∞ β ∫ 0 T b ( t ) d t − a 0 ∫ 0 T b ( t ) d t ≥ 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) − C ‖ u ˙ ‖ L 2 β − C .

According to ( H 5 ) , there exists a constant a 1 > T 2 2 π 2 > 0 satisfying

(2.5) lim ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) 1 μ ∫ 0 T F 1 ( t , λ x ) d t + ∫ 0 T F 2 ( t , x ) d t > C 0 2 a 1 4 ‖ f ‖ L 2 2 .

It follows from ( H 3 ) with α ∈ [ 0 , 1 ) Hölder inequality and Lemma 2.1 that

(2.6) ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t = ∫ 0 T ∫ 0 1 ( ∇ F 2 ( t , u ¯ + s u ˜ ( t ) ) , u ˜ ( t ) ) d s d t ≤ ∫ 0 T ∫ 0 1 ( f ( t ) h ( ∣ u ¯ + s u ˜ ( t ) ∣ ) + g ( t ) ) ∣ u ˜ ( t ) ∣ d s d t ≤ ∫ 0 T ∫ 0 1 C 0 f ( t ) ( h ( ∣ u ¯ ∣ ) + h ( ‖ u ˜ ‖ ∞ ) ) ∣ u ˜ ( t ) ∣ d s d t + ‖ u ˜ ‖ ∞ ∫ 0 T g ( t ) d t ≤ C 0 h ( ∣ u ¯ ∣ ) ∫ 0 T f ( t ) ∣ u ˜ ( t ) ∣ d t + C 0 K 1 ‖ u ˜ ‖ ∞ α + 1 ∫ 0 T f ( t ) d t + ‖ u ˜ ‖ ∞ C 0 K 2 ∫ 0 T f ( t ) d t + ∫ 0 T g ( t ) d t ≤ C 0 h ( ∣ u ¯ ∣ ) ∫ 0 T ∣ f ( t ) ∣ 2 d t 1 2 ∫ 0 T ∣ u ˜ ( t ) ∣ 2 d t 1 2 + C ‖ u ˙ ‖ L 2 α + 1 + C ‖ u ˙ ‖ L 2 ≤ C 0 1 C 0 a 1 ‖ u ˜ ‖ L 2 2 + C 0 a 1 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + C ‖ u ˙ ‖ L 2 α + 1 + C ‖ u ˙ ‖ L 2 ≤ T 2 4 a 1 π 2 ‖ u ˙ ‖ L 2 2 + C 0 2 a 1 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + C ‖ u ˙ ‖ L 2 α + 1 + C ‖ u ˙ ‖ L 2 .

Combining (2.4), (2.6), and the definition of φ in (1.2), we obtain

(2.7) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 1 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 1 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 − C ‖ u ˙ ‖ L 2 α + 1 − C ‖ u ˙ ‖ L 2 + 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t − C ‖ u ˙ ‖ L 2 β − C = 1 2 − T 2 4 a 1 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 α + 1 − C ‖ u ˙ ‖ L 2 − C ‖ u ˙ ‖ L 2 β − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t − C 0 2 a 1 4 ‖ f ‖ L 2 2 .

Since ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , combining (2.5), (2.7), and (iv) of ( H α ) with α ∈ [ 0 , 1 ) , we have φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . Therefore, by Lemma 2.2, we prove that Case 1 holds.

Proof of Case 2. By ( H 5 * ) and ( f 1 ) , there exists a constant a 2 > T 2 2 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ > 0 satisfying

(2.8) lim ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) 1 μ ∫ 0 T F 1 ( t , λ x ) d t + ∫ 0 T F 2 ( t , x ) d t > C 0 2 a 2 4 ‖ f ‖ L 2 2 .

Combining ( H 3 ) with α = 1 and Lemma 2.1, we have

(2.9) ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t = ∫ 0 T ∫ 0 1 ( ∇ F 2 ( t , u ¯ + s u ˜ ( t ) ) , u ˜ ( t ) ) d s d t ≤ ∫ 0 T ∫ 0 1 ( f ( t ) h ( ∣ u ¯ + s u ˜ ( t ) ∣ ) + g ( t ) ) ∣ u ˜ ( t ) ∣ d s d t ≤ ∫ 0 T ∫ 0 1 C 0 f ( t ) ( h ( ∣ u ¯ ∣ ) + h ( ∣ u ˜ ( t ) ∣ ) ) ∣ u ˜ ( t ) ∣ d s d t + ‖ u ˜ ‖ ∞ ∫ 0 T g ( t ) d t ≤ C 0 h ( ∣ u ¯ ∣ ) ∫ 0 T f ( t ) ∣ u ˜ ( t ) ∣ d t + C 0 K 1 ∫ 0 T f ( t ) ∣ u ˜ ( t ) ∣ 2 d t + ‖ u ˜ ‖ ∞ C 0 K 2 ∫ 0 T f ( t ) d t + ∫ 0 T g ( t ) d t ≤ C 0 h ( ∣ u ¯ ∣ ) ∫ 0 T ∣ f ( t ) ∣ 2 d t 1 2 ∫ 0 T ∣ u ˜ ( t ) ∣ 2 d t 1 2 + C 0 K 1 ‖ f ‖ ∞ ∫ 0 T ∣ u ˜ ( t ) ∣ 2 d t + C ‖ u ˙ ‖ L 2 ≤ C 0 1 C 0 a 2 ‖ u ˜ ‖ L 2 2 + C 0 a 2 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 + C ‖ u ˙ ‖ L 2 ≤ T 2 4 a 2 π 2 + C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 + C 0 2 a 2 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + C ‖ u ˙ ‖ L 2 .

From (2.4), (2.9), and (1.2), we obtain

(2.10) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 2 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 2 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 − C ‖ u ˙ ‖ L 2 + 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t − C ‖ u ˙ ‖ L 2 β − C = 1 2 − T 2 4 a 2 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 − C ‖ u ˙ ‖ L 2 β − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) 1 μ ∫ 0 T F 1 ( t , λ u ¯ ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t − C 0 2 a 2 4 ‖ f ‖ L 2 2 .

Since ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , so by (2.8), (2.10) and (iv) of ( H α ) with α = 1 , we obtain φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . Thus, by Lemma 2.2, we complete the proof of Case 2. Hence, we complete the proof of Theorem 1.1.□

Proof of Theorem 1.2

We should divide the proof into two cases.

Proof of Case 1. Let β = log 2 λ ( 2 μ ) , then from ( H 1 * ) , we have 0 ≤ β < 2 . For ∣ x ∣ > 1 , there exists positive integer n satisfying

− n ≤ log 2 λ ∣ x ∣ < − ( n − 1 ) .

Then, we obtain ∣ x ∣ β ≥ 1 ( 2 λ ) ( n − 1 ) β = 1 ( 2 μ ) n − 1 and ∣ x ∣ ≤ 1 ( 2 λ ) n . It follows from conditions ( H 1 * ) and ( A ) that

F 1 ( t , x ) ≥ 1 2 μ F 1 ( t , 2 λ x ) ≥ ⋯ ≥ 1 2 μ 1 ( 2 μ ) ( n − 1 ) F 1 ( t , ( 2 λ ) n x ) ≥ − ∣ x ∣ β a 0 b ( t ) 2 μ

for a.e. t ∈ [ 0 , T ] and each ∣ x ∣ ≥ 1 , where a 0 = max 0 ≤ s ≤ 1 a ( s ) . What’s more, one obtain

(2.11) F 1 ( t , x ) ≥ − 1 2 μ ∣ x ∣ β + 1 a 0 b ( t ) ,

for a.e. t ∈ [ 0 , T ] and each x ∈ R N .

As F 1 ( t , ⋅ ) is ( λ , μ ) -subconvex, then we have

F 1 ( t , λ u ˜ ( t ) ) ≤ μ ( F 1 ( t , u ( t ) ) + F 1 ( t , − u ¯ ) ) ,

i.e.,

(2.12) ∫ 0 T F 1 ( t , u ( t ) ) d t ≥ 1 μ ∫ 0 T F 1 ( t , λ u ˜ ( t ) ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t .

Hence, combining (2.11), (2.12), and Sobolev inequality, we obtain

(2.13) ∫ 0 T F 1 ( t , u ( t ) ) d t ≥ 1 μ ∫ 0 T F 1 ( t , λ u ˜ ( t ) ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t ≥ − 1 μ ∫ 0 T 1 2 μ ∣ u ˜ ( t ) ∣ β + 1 a 0 b ( t ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t ≥ − 1 μ 1 2 μ ‖ u ˜ ‖ ∞ β + 1 ∫ 0 T a 0 b ( t ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t ≥ − C ‖ u ˙ ‖ L 2 β − C − ∫ 0 T F 1 ( t , − u ¯ ) d t .

According to ( H 6 ) , there is a constant a 3 > T 2 2 π 2 > 0 satisfying

(2.14) lim ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t − ∫ 0 T F 1 ( t , − x ) d t > C 0 2 a 3 4 ‖ f ‖ L 2 2 .

By (2.6), where a 3 instead of a 1 , (2.13), and (1.2), we have

(2.15) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 3 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 3 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 − C ‖ u ˙ ‖ L 2 α + 1 + ∫ 0 T F 2 ( t , u ¯ ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t − C ‖ u ˙ ‖ L 2 − C ‖ u ˙ ‖ L 2 β − C = 1 2 − T 2 4 a 3 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 α + 1 − C ‖ u ˙ ‖ L 2 − C ‖ u ˙ ‖ L 2 β − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) ∫ 0 T F 2 ( t , u ¯ ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t − C 0 2 a 3 4 ‖ f ‖ L 2 2 .

Since ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , from (2.14), (2.15), and (iv) of ( H α ) with α ∈ [ 0 , 1 ) , we derive φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . According to Lemma 2.2, we complete the proof of Case 1.

Proof of Case 2. By ( H 6 * ) and ( f 1 ) , there is a constant a 4 > T 2 2 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ > 0 so that

(2.16) lim ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t − ∫ 0 T F 1 ( t , − x ) d t > C 0 2 a 4 4 ‖ f ‖ L 2 2 .

It follows from (2.9), where a 4 instead of a 2 , (2.13), and (1.2) that

(2.17) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 4 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 4 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + ∫ 0 T F 2 ( t , u ¯ ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t − C ‖ u ˙ ‖ L 2 − C ‖ u ˙ ‖ L 2 β − C = 1 2 − T 2 4 a 4 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 − C ‖ u ˙ ‖ L 2 β − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) ∫ 0 T F 2 ( t , u ¯ ) d t − ∫ 0 T F 1 ( t , − u ¯ ) d t − C 0 2 a 4 4 ‖ f ‖ L 2 2 .

As ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , then combining (2.16), (2.17), and (iv) of ( H α ) with α = 1 , we obtain φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . By Lemma 2.2, we prove that Case 2 holds. Therefore, we complete the proof of Theorem 1.2.□

Proof of Theorem 1.3

We have three cases to prove.

Proof of Case 1. From ( H 2 ) , one obtain

(2.18) ∫ 0 T F 1 ( t , u ( t ) ) d t ≥ ∫ 0 T ( ξ ( t ) , u ¯ + u ˜ ( t ) ) d t + ∫ 0 T γ ( t ) d t = ∫ 0 T ( ξ ( t ) , u ˜ ( t ) ) d t + ∫ 0 T γ ( t ) d t ≥ − ‖ u ˜ ‖ ∞ ∫ 0 T ∣ ξ ( t ) ∣ d t + ∫ 0 T γ ( t ) d t ≥ − C ‖ u ˙ ‖ L 2 − C .

By ( H 4 ) , there is a constant a 5 > T 2 2 π 2 > 0 so that

(2.19) lim ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t > C 0 2 a 5 4 ‖ f ‖ L 2 2 .

Then, combining (2.6), where a 5 instead of a 1 , (2.18), and (1.2), we have

(2.20) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 5 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 5 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + ∫ 0 T F 2 ( t , u ¯ ) d t − C ‖ u ˙ ‖ L 2 α + 1 − C ‖ u ˙ ‖ L 2 − C = 1 2 − T 2 4 a 5 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 α + 1 − C ‖ u ˙ ‖ L 2 − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) ∫ 0 T F 2 ( t , u ¯ ) d t − C 0 2 a 5 4 ‖ f ‖ L 2 2 .

Since ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , it follows from (2.19), (2.20), and (iv) of ( H α ) with α ∈ [ 0 , 1 ) that φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . It follows from Lemma 2.1 that Case 1 holds.

Proof of Case 2. By ( H 4 * ) and ( f 1 ) , there exists a constant a 6 > T 2 2 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ > 0 satisfying

(2.21) lim ∣ x ∣ → + ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t > C 0 2 a 6 4 ‖ f ‖ L 2 2 .

From (2.9), where a 6 instead of a 2 , (2.18), and (1.2), we have

(2.22) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 6 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 6 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + ∫ 0 T F 2 ( t , u ¯ ) d t − C ‖ u ˙ ‖ L 2 − C = 1 2 − T 2 4 a 6 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) ∫ 0 T F 2 ( t , u ¯ ) d t − C 0 2 a 6 4 ‖ f ‖ L 2 2 .

Since ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , hence according to (2.21), (2.22), and (iv) of ( H α ) with α = 1 , we obtain φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . From Lemma 2.2, we complete the proof of Case 2.

Proof of Case 3. It follows from ( H 2 * ) that

(2.23) ∫ 0 T F 1 ( t , u ( t ) ) d t ≥ ∫ 0 T ( ξ ( t ) , u ¯ + u ˜ ( t ) ) d t + ∫ 0 T γ ( t ) d t ≥ − ∫ 0 T ∣ ξ ( t ) ∣ ∣ u ˜ ( t ) ∣ d t − ∫ 0 T ∣ ξ ( t ) ∣ ∣ u ¯ ∣ d t + ∫ 0 T γ ( t ) d t ≥ − ‖ u ˜ ‖ ∞ ∫ 0 T ∣ ξ ( t ) ∣ d t − ∣ u ¯ ∣ ∫ 0 T ∣ ξ ( t ) ∣ d t + ∫ 0 T γ ( t ) d t ≥ − C ‖ u ˙ ‖ L 2 − C ∣ u ¯ ∣ − C .

From ( H 4 * ) and ( f 1 ) , there is a constant a 7 > T 2 2 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ > 0 such that

(2.24) lim ∣ x ∣ → ∞ 1 h 2 ( ∣ x ∣ ) ∫ 0 T F 2 ( t , x ) d t > C 0 2 a 7 4 ‖ f ‖ L 2 2 .

Combining (2.9), where a 7 instead of a 2 , (2.23), and (1.2), we obtain

(2.25) φ ( u ) = 1 2 ‖ u ˙ ‖ L 2 2 + ∫ 0 T F 1 ( t , u ( t ) ) d t + ∫ 0 T ( F 2 ( t , u ( t ) ) − F 2 ( t , u ¯ ) ) d t + ∫ 0 T F 2 ( t , u ¯ ) d t ≥ 1 2 − T 2 4 a 7 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C 0 2 a 7 4 h 2 ( ∣ u ¯ ∣ ) ‖ f ‖ L 2 2 + ∫ 0 T F 2 ( t , u ¯ ) d t − C ‖ u ˙ ‖ L 2 − C ∣ u ¯ ∣ − C = 1 2 − T 2 4 a 7 π 2 − C 0 K 1 T 2 ‖ f ‖ ∞ 4 π 2 ‖ u ˙ ‖ L 2 2 − C ‖ u ˙ ‖ L 2 − C + h 2 ( ∣ u ¯ ∣ ) 1 h 2 ( ∣ u ¯ ∣ ) ∫ 0 T F 2 ( t , u ¯ ) d t − C ∣ u ¯ ∣ h 2 ( ∣ u ¯ ∣ ) − C 0 2 a 7 4 ‖ f ‖ L 2 2 .

Since ‖ u ‖ → + ∞ ⇔ ( ∣ u ¯ ∣ 2 + ‖ u ˙ ‖ L 2 2 ) 1 2 → + ∞ , so by (2.24), (2.25), and ( H 3 * ) , we obtain φ ( u ) → + ∞ as ‖ u ‖ → + ∞ . By Lemma 2.2, we infer that Case 3 holds. Hence, the proof of Theorem 1.3 is complete.□

Proof of Theorem 1.4

First, we check that functional φ satisfies (PS) condition. In fact, assume sequence { u n } is a (PS) sequence, i.e., { φ ( u n ) } is bounded and φ ′ ( u n ) → 0 as n → + ∞ , then from the proof of Theorem 1.3, we see that φ satisfies the coercive condition, which means that { u n } is bounded. Since H T 1 is a reflexive Banach space, u n ⇀ u in H T 1 in subsequence sense. From Sobelev’s embedding theorem, we obtain u n → u in C ( [ 0 , T ] , R N ) . Then, we obtain

∫ 0 T ∣ u ˙ n ( t ) − u ˙ ( t ) ∣ 2 d t = ⟨ φ ′ ( u n ) − φ ′ ( u ) , u n − u ⟩ − ∫ 0 T ( ∇ F ( t , u n ( t ) ) − ∇ F ( t , u ( t ) ) , u n ( t ) − u ( t ) ) d t → 0 , as n → + ∞ ,

which means ‖ u ˙ n − u ˙ ‖ L 2 → 0 . Then, we have u n → u in H T 1 . So we prove that functional φ satisfies (PS) condition.

Next, we will verify that condition (2.1) of Lemma 2.3 holds. Let X = H T 1 , and write X = X 1 ⊕ X 2 , X 2 = X 1 ⊥ with

X 2 = ∑ j = 0 k ( a j cos j ω t + b j sin j ω t ) ∣ a j , b j ∈ R N , j = 0 , … , k .

Then, we have

∫ 0 T ∣ u ( t ) ∣ 2 d t = T ∑ j = 0 k a j 2 + b j 2 2 , ∫ 0 T ∣ u ˙ ( t ) ∣ 2 d t = T ω 2 ∑ j = 0 k j 2 a j 2 + b j 2 2 , ∀ u ∈ X 2 .

According to Sobelev’s embedding theorem, there holds ‖ u ‖ ∞ ≤ C ‖ u ‖ . By ( H 7 ) , we have

(2.26) φ ( u ) ≤ 1 2 ∫ 0 T ∣ u ˙ ( t ) ∣ 2 d t − 1 2 k 2 ω 2 ∫ 0 T ∣ u ( t ) ∣ 2 d t ≤ 0 , ∀ u ∈ X 2 , ‖ u ‖ ≤ r C .

In a same way, we obtain

φ ( u ) ≥ 1 2 ∫ 0 T ∣ u ˙ ( t ) ∣ 2 d t − 1 2 ( k + 1 ) 2 ω 2 ∫ 0 T ∣ u ( t ) ∣ 2 d t ≥ 0 , ∀ u ∈ X 1 , ‖ u ‖ ≤ r C .

Now, we have two cases.

Case 1. If inf X φ < 0 , by Lemma 2.3, we prove that functional φ possesses at least three critical points in H T 1 .

Case 2. If inf X φ ≥ 0 , according to (2.26), we obtain φ ( u ) = 0 for any u ∈ X 2 with ‖ u ‖ ≤ r C . Thus, functional φ has infinitely many critical points.

Hence, we complete the proof of Theorem 1.4.□

Acknowledgement

The authors thank the referees for their valuable suggestions and comments on the manuscript.

  1. Funding information: Chungen Liu was partially supported by the NSF of China (12171108).

  2. Author contributions: The two authors contributed equally to this work.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] N. Aizmahin and T. Q. An, The existence of periodic solutions of non-autonomous second-order Hamiltonian systems, Nonlinear Anal. 74 (2011), no. 14, 4862–4867, DOI: https://doi.org/10.1016/j.na.2011.04.060. 10.1016/j.na.2011.04.060Search in Google Scholar

[2] M. S. Berger and M. Schechter, On the solvability of semilinear gradient operator equations, Adv. Math. 5 (1977), no. 2, 97–132, DOI: https://doi.org/10.1016/0001-8708(77)90001-9. 10.1016/0001-8708(77)90001-9Search in Google Scholar

[3] H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8–9, 939–963. 10.1002/cpa.3160440808Search in Google Scholar

[4] X. F. Chen and F. Guo, Existence and multiplicity of periodic solutions for nonautonomous second-order Hamiltonian systems, Bound. Value Probl. 138 (2016), 1–10, DOI: http://doi.org/10.1186/s13661-016-0647-y. 10.1186/s13661-016-0647-ySearch in Google Scholar

[5] Y. M. Long, Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear Anal. 24 (1995), no. 12, 1665–1671, DOI: https://doi.org/10.1016/0362-546X(94)00227-9. 10.1016/0362-546X(94)00227-9Search in Google Scholar

[6] X. Lv, S. P. Lu, and P. Yan, Periodic solutions of non-autonomous ordinary p-Laplacian systems, J. Appl. Math. Comput. 35 (2011), no. 1–2, 11–18, DOI: https://doi.org/10.1007/s12190-009-0336-4. 10.1007/s12190-009-0336-4Search in Google Scholar

[7] J. Ma and C. L. Tang, Periodic solutions for some nonautonomous second-order systems, J. Math. Anal. Appl. 275 (2002), no. 2, 482–494, DOI: https://doi.org/S0022-247X(02)00636-4. 10.1016/S0022-247X(02)00636-4Search in Google Scholar

[8] J. Mawhin, Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci. 73 (1987), no. 3–4, 118–130, DOI: https://doi.org/10.3406/barb.1987.57663. 10.3406/barb.1987.57663Search in Google Scholar

[9] J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer, New York, 1989, DOI: https://doi.org/10.1007/978-1-4757-2061-7. 10.1007/978-1-4757-2061-7Search in Google Scholar

[10] F. J. Meng, Existence of periodic solutions for a class of second-order Hamiltonian system, Jiangxi Sci. 27 (2009), no. 5, 654–656. Search in Google Scholar

[11] F. J. Meng, Existence of periodic solutions for a class of second-order Hamiltonian system, Sci. Tech. Eng. 9 (2009), no. 21, 6286–6288. Search in Google Scholar

[12] Q. Meng and X. H. Tang, Solutions of a second-order Hamiltonian system with periodic boundary conditions, Commun. Pure Appl. Anal. 9 (2010), no. 4, 1053–1067, DOI: https://doi.org.10.3934/cpaa.2010.9.1053. 10.3934/cpaa.2010.9.1053Search in Google Scholar

[13] P. H. Rabinowitz, On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 33 (1980), no. 5, 609–633, DOI: https://doi.org/10.1002/cpa.3160330504. 10.1002/cpa.3160330504Search in Google Scholar

[14] Y. F. Ru and F. L. Wang, Existence of nonconstant periodic solutions for p(t)-Laplacian Hamiltonian system, Bound. Value Probl. 135 (2019), 1–13, DOI: https://doi.org/10.1186/s13661-019-1248-3. 10.1186/s13661-019-1248-3Search in Google Scholar

[15] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Third edition, Springer-Verlag, Berlin, 2000, DOI: https://doi.org/10.1007/978-3-540-74013-1. 10.1007/978-3-540-74013-1Search in Google Scholar

[16] H. M. Suo, L. Di, Y. C. An, and C. M. Chu, Existence and multiplicity of periodic solutions for some second-order Hamiltonian systems, J. Inequal. Appl. 411 (2014), 1–13, DOI: https://doi.org/10.1186/1029-242X-2014-411. 10.1186/1029-242X-2014-411Search in Google Scholar

[17] C. L. Tang, Periodic solutions of non-autonomous second-order systems with γ-quasi sub additive potential, J. Math. Anal. Appl. 189 (1995), no. 3, 671–675, DOI: https://doi.org/10.1006/jmaa.1995.1044. 10.1006/jmaa.1995.1044Search in Google Scholar

[18] C. L. Tang, Existence and multiplicity of periodic solutions for nonautonomous second-order systems, Nonlinear Anal. 32 (1998), no. 3, 299–304, DOI: https://doi.org/10.1016/S0362-546X(97)00493-8. 10.1016/S0362-546X(97)00493-8Search in Google Scholar

[19] C. L. Tang, Periodic solutions for nonautonomous second-order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126 (1998), no. 11, 3263–3270, DOI: https://www.jstor.org/stable/119141. 10.1090/S0002-9939-98-04706-6Search in Google Scholar

[20] X. H. Tang and Q. Meng, Solutions of a second-order Hamiltonian system with periodic boundary conditions, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3722–3733, DOI: https://doi.org/10.1016/j.nonrwa.2010.02.002. 10.1016/j.nonrwa.2010.02.002Search in Google Scholar

[21] X. H. Tang and X. Y. Zhang, Periodic solutions for second-order Hamiltonian systems with a p-Laplacian, Ann. Univ. Mariae Curie-Skllllodowska Sect. A 64 (2010), no. 1, 93–113, DOI: https://doi.org/10.2478/v10062-010-0008-8. 10.2478/v10062-010-0008-8Search in Google Scholar

[22] X. P. Wu and C. L. Tang, Periodic solutions of a class of non-autonomous second-order systems, J. Math. Anal. Appl. 236 (1999), no. 2, 227–235, DOI: https://doi.org/10.1006/jmaa.19.99.6408. 10.1006/jmaa.1999.6408Search in Google Scholar

[23] Z. Y. Wang and J. H. Zhang, Periodic solutions of a class of second-order non-autonomous Hamiltonian systems, Nonlinear Anal. 72 (2010), no. 12, 4480–4487, DOI: https://doi.org/10.1016/j.na.2010.02.023. 10.1016/j.na.2010.02.023Search in Google Scholar

[24] Z. Y. Wang and J. H. Zhang, Periodic solutions for nonautonomous second-order Hamiltonian systems with sublinear nonlinearity, Bound. Value Probl. 23 (2011), 1–14, DOI: https://doi.org/10.1186/1687-2770-2011-23. 10.1186/1687-2770-2011-23Search in Google Scholar

[25] F. K. Zhao and X. Wu, Periodic solutions for a class of non-autonomous second-order systems, J. Math. Anal. Appl. 296 (2004), no. 2, 422–434, DOI: https://doi.org/10.1016/j.jmaa.2004.01.041. 10.1016/j.jmaa.2004.01.041Search in Google Scholar

[26] F. K. Zhao and X. Wu, Existence and multiplicity of periodic solution for nonautonomous second-order systems with linear nonlinearity, Nonlinear Anal. 60 (2005), no. 2, 325–335, DOI: https://doi.org/10.1016/j.na.2004.08.031. 10.1016/S0362-546X(04)00348-7Search in Google Scholar

[27] Q. F. Zhang and X. H. Tang, Periodic solutions for second-order Hamiltonian systems, Appl. Math. 57 (2012), no. 4, 407–425, DOI: https://doi.org/10.1007/s10492-012-0024-9. 10.1007/s10492-012-0024-9Search in Google Scholar

[28] L. Zhang and X. H. Tang, Periodic solutions for some nonautonomous p(t)-Laplacian Hamiltonian systems, Appl. Math. 58 (2013), no. 1, 39–61, DOI: https://doi.org/10.1007/s10492-013-0002-x. 10.1007/s10492-013-0002-xSearch in Google Scholar

[29] S. G. Zhang, Periodic solutions for a class of second-order Hamiltonian systems with p(t)-Laplacian, Bound. Value Probl. 211 (2016), 1–20, DOI: https://doi.org/10.1186/s13661-016-0720-6. 10.1186/s13661-016-0720-6Search in Google Scholar
Received: 2023-07-04
Revised: 2024-09-05
Accepted: 2024-10-14
Published Online: 2024-12-31

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
https://doi.org/10.1515/anona-2024-0051
Keywords for this article
periodic solutions; the least action principle; minimax methods; second-order Hamiltonian systems
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 7.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anona-2024-0051/html
Scroll to top button