Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

. In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for


Introduction and main results
In this paper, we consider the following second-order planar Hamiltonian systems ü(t) + ∇V(t, u(t)) = 0, (1.1) where V : R × R 2 → R is a C 1 -map.We say that a solution u(t) of problem (1.1) is nontrivial homoclinic (to 0) if u ̸ ≡ 0, u(t) → 0 and u(t) → 0 as t → ±∞.Subsequently, ∇V(t, x) denotes the gradient with respect to the x variable, (•, •) : R 2 × R 2 → R denotes the standard inner product in R 2 and | • | is the induced norm.
Hamiltonian system is a classical model in celestial mechanics, fluid mechanics and so on.Since its importance in physic fields, searching for the solutions of the Hamiltonian systems has attracted much attention of mathematicians since Poincaré.In a remarkable paper [31], the periodic solutions are firstly obtained for (1.1) with prescribed energy and prescribed period cases respectively via variational methods by Rabinowitz.However, to show homoclinic solutions via variational methods seems difficult since the lack of compactness for Email: wudl2008@163.com(DL) W(t, x) > 0 if x ̸ = 0, and there exist ϵ ∈ (0, 1) and c > 0 such that W(t, x) ≥ c (∇W(t, x), x) |x| 2−ϵ for |x| large enough.
In this paper, we mainly consider the asymptotically autonomous potentials without periodic, coercive, even assumption or perturbations.In 1999, Carrião and Miyagaki [5] showed the existence of homoclinic for problem (1.1) by assuming that V(t, x) converges to V ∞ (x) as |t| → +∞ and V ∞ (x) satisfies the (AR) condition.The asymptotically autonomous Hamiltonian systems has also been considered by Lv, Xue and Tang [26] with asymptotically quadratic potentials.They showed the existence of homoclinic solutions for systems (1.1) with a(t) ≡ const.being small enough.In another paper, Lv, etc. [27] also obtained ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems.Their results generalized the conclusions in [1,5] by replacing the (AR) condition with strict monotonic conditions on W(t, x).
In this paper, we mainly consider the combined nonlinearities.In [6,26,36,37,44,46], the authors also considered the following concave-convex potentials where a(t) is coercive, i.e. a(t) → +∞ as t → ∞, F(t, x) is subquadratic and G(t, x) is superquadratic in x ∈ R N .The coercivity of a(t) is an important assumption which can guarantee the compactness of Sobolev embedding.
In [46], Yang, Chen and Sun assumed that a(t) is coercive, F(t, x) = m(t)|x| γ and G(t, x) = d|x| p with m ∈ L 2 2−γ (R, R + ) and 1 < γ < 2, d ≥ 0, p > 2. This result is generalized by Chen and He [6] with the following generalized superquadratic condition (CH) There exist ρ > 2 and 1 < δ < 2 such that Obviously, (CH) is weaker than (AR) since h(t) > 0 for all t ∈ R. In [42], Wu, Tang and Wu generalized the above results by relaxing the conditions on G.However, a(t) is also required to be coercive.
Without coercive assumption, there are also some other papers concerning on this case with the steep well potentials (see [36,37]).In [36], the nonlinearities are the combination of subquadratic and asymptotic quadratic nonlinearities.While in [37], the nonlinearities are the combination of superquadratic and subquadratic nonlinearities.In [46], Ye and Tang obtained infinitely many homoclinic solutions for systems (1.1) with where a(t) ≥ 0 and By assuming h(t) > 0, the authors in [46] constructed a sequence of negative critical values.However, in [5,26,27], W(t, x) is assumed to be non-negative in R × R N .A natural question is whether (1.1) possesses homoclinic solutions if W(t, x) change signs without periodic or coercive assumptions.In this paper, we partially give some answers to this question.Precisely, we consider the sign-changing and asymptotically autonomous potentials, which have not been considered before as we know.Hence, we cannot obtain our results as the authors did in [6,26,36,37,44,46].Concentration-compactness principle(CCP) is adopted to show the compactness.The crucial step in using the (CCP) is to exclude the dichotomy case by estimating the critical values.This can be easily done if W satisfies the following monotonic condition (MC) the mapping τ → ∇W(t,τx) τ , x is strictly increasing in τ ∈ (0, 1] for all x ̸ = 0 and t ∈ R. However, condition (MC) is not valid for our potentials.Hence we need more delicate estimates for the critical values to show the contradictions.The constant for the Sobolev inequality plays an important role in obtaining our results.In the next section, we show the best constant for the Sobolev inequality.

Best constant for the Sobolev inequality
Let's make it clear that L p (R, R m ) and H 1 (R, R m ) are the Banach spaces of functions on R valued in R m under the norms where C ν is the best constant which is defined in the following proof.This inequality is important in using variational methods to show the existence and multiplicity of differential equations.However, since the best constant for the Sobolev inequality seems not important in previous studies of Hamiltonian systems, as we know, there is no paper concerning on the best constant of Sobolev inequality for (2.1).In this section, we show the best constant for (2.1).
There have been many papers concerning on the best constant for the Sobolev inequality in H 1 (R, R) (see [2][3][4]12]).In a remarkable paper, Talenti [38] obtained the best constant for Sobolev inequality in H 1 (R N , R) with N > 1.In 1983, Weinstein obtained the best constant for the following Gagliardo-Nirenberg-Sobolev inequalities where is the best constant for (2.2) and G is the unique positive solution for the following scalar field equation In a recent paper, Dolbeault, etc. [10] considered the best constant for the one-dimensional Gagliardo-Nirenberg-Sobolev inequalities in H 1 (R, R m )(m = 1) and obtained where M GN (ν) is defined as Moreover, M GN (ν) is attained at v ⋆ , which is the unique optimal function up to translations, multiplication by a constant and scalings, defined as The following computation is made by the authors in [10].For the reader's convenience, we write them here.
Subsequently, we consider the case m > 1.For any u(t) = (u Then we have For any ν > 2, let On one hand, if we choose u 1 = . . .= u m , it is easy to see that R ≤ 1.On the other hand, we can also deduce that R ≥ 1 since where M GN (ν) is the best constant defined in (2.4) and attained at and M GN (ν) is the best constant which can be attained if and only if . (2.11) It follows from (2.10) and (2.11) that inf . (2.13) Then, we infer that (2.1) holds and is the best constant.Moreover, we also need to consider the best constant when ν = +∞.

Solutions for the limit systems
In this section, we consider the solutions for the limit systems of (1.1).In the rest of this paper, we only consider the systems in R 2 .The potential V is defined as where a, d ∈ C(R, R), λ > 0, ν > 2 and the following conditions hold (V1) there exists a 0 > 0 such that a(t) ≥ a 0 for all t ∈ R; Here Q ν : R → R + is the unique positive ground state solution(up to translations) for the following equation Let us consider the following systems The existence of solutions for systems (3.2) has been considered by many mathematicians via the variational methods.A solution (u 1 , . . ., u m ) for (3.2) is said to be positive if u 1 , . . ., u m > 0.
When m = 1, (3.2) reduces to a differential equation and the uniqueness of positive ground state solution for (3.2) has been shown by M. K. Kwong [19] with ν > 2. The readers are also referred to [17,34] for more general cases.When m > 1, (3.2) is related to the coupled nonlinear Schrödinger equations.In last decades, there have been many mathematicians devoting themselves to the uniqueness of positive solutions for the coupled nonlinear Schrödinger equations and obtained many significant results (see [9,19,26,41]).In a recent paper [41], Wei and Yao considered the following systems they showed the uniqueness of positive solutions for system (3.3),defined as where w 0 is the unique positive solution of q When λ 1 = λ 2 = λ and µ 1 = µ 2 = β, it has also been shown in [41] that all the positive solutions of system (3.3) have the following form For the high dimension cases, i.e. n = 2, 3 and m = 2, the readers are referred to another paper by Dai, Tian and Zhang [11].However, the case n = 1 is not considered.We can see that (3.2) reduces to (3.3) if ν = 4. Motivated by above papers, we obtain the uniqueness of solutions for (3.2) when m = 2, n = 1 and ν > 2.More precisely, we obtain the following lemma.
Then system (3.2) possesses at least one positive solution.Let U ν : R → R + × R + be a positive solution for systems (3.2), then there exits ω ∈ (0, π/2) such that and U ν is the ground state solution for (3.2).
Proof.Since n = 1, the critical exponent equals to +∞.The existence of positive solutions for the subcritical problems have been considered in [7,13,35,41].Subsequently, we only show (3.4) holds and U ν is the ground state solution for (3.2).Let Then M ν = (M 1 (t), M 2 (t)) is the positive solution for the following system By the ordinary differential equation theory, one can deduce Combining (3.6) and (3.7), we obtain , we see T(t) > 0 satisfies (3.1).By the uniqueness, one has which implies (3.4).We also show that U ν is a ground state solution for systems (3.2).Actually, the corresponding functional of (3.2) is defined as . Moreover, the corresponding functional of (3.1) is defined as Obviously, for any q(t) ∈ N and e ∈ R 2 with |e| = 1, we have that In turn, for any and Remark 3.2.When ν = 4, Theorem 3.1 reduces to the results in [41].

Main results
In this section, we prove our main result.
Remark 4.2.In [36,37], Sun and Wu also considered (1.1) with mixed nonlinearities.In both papers, the infimum of a(t) cannot be attained at infinity, which is different from our result.
Remark 4.3.In Theorem 4.1, there are no periodic, coercive or symmetric assumptions on a(t), which is different from the results in [6,14,32,39,44].According to our conditions, both of the superquadratic and subquadratic parts of V can change signs, then we can not obtain the compactness as the authors did in [46].
Remark 4.4.In [26,27], W(t, x) is required to satisfy and (∇W(t, x), x) ≥ 2W(x) for all t ∈ R, x ∈ R N , ( where W ∞ is the limit function of W as t → ∞.In our theorem, we have Since F(t, x) and d(t) can change signs, we infer that (4.1) and (4.2) are not valid for (4.3).Moreover, since (4.1), (4.2) and (MC) hold in [27], the authors can show that for any u ∈ H 1 (R, R 2 ), there exists unique s u > 0 such that s u u ∈ L and sup s≥0 I(su) = I(s u u), where This conclusion is crucial in using the (CCP) to show the contradictions.However, we can not obtain this conclusion by our conditions.Therefore, the Nehari-manifold method is not applicable for our theorem.

Preliminaries
The corresponding functional of (1.1) is defined by Proof.The proof is similar to Lemma 2.3 in [6].
Lemma 4.6.The critical points of I are homoclinic solutions for problem (1.1).
We will show the existence of two critical points of I by the Mountain Pass Theorem and the following critical point lemma respectively.Lemma 4.7 (Lu [22]).Let X be a real reflexive Banach space and Ω ⊂ X be a closed bounded convex subset of X. Suppose that φ : X → R is a weakly lower semi-continuous (w.l.s.c. for short) functional.If there exists a point x 0 ∈ Ω \ ∂Ω such that φ(x) > φ(x 0 ), ∀ x ∈ ∂Ω, then there must be an x * ∈ Ω \ ∂Ω such that

The Mountain Pass Structure
In this section, we mainly show the Mountain Pass structure of I and obtain some crucial estimates.
By the Mountain Pass theorem, there exists a sequence {u n } and c ≥ α given by where and for any v Next, we show an important relation between c and c ∞ , which is crucial in the following concentration compactness study.
First, we show the boundedness of ∥u n ∥.It follows from (4.4), (4.5), (4.7), (4.8) and (4.10) that Hence there exists D > 0 such that ∥u n ∥ ≤ D for all n ∈ N. (4.11)Without loss of generality, we assume that On one hand, we can easily deduce that On the other hand, it can be easily deduce from (4.15) that and Then one has Furthermore, we can deduce that The discussion for this step is divided into two cases.
First, we show the following claim.
Claim 1: By the arbitrariness of ε, we can see that we deduce that A n → 0 and σ n → 1 as n → ∞.Setting z n = σ n w n (t), we have = 0, which implies z n ∈ N .Furthermore, we have which implies