Ground state homoclinic orbits of damped vibration problems

In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case.

MSC: 49J40, 70H05.

We shall study the existence of ground state homoclinic orbits for the following non-periodic damped vibration system:

where M is an antisymmetric N\times N constant matrix, L(t)\in C(\mathbb{R},{\mathbb{R}}^{N\times N}) is a symmetric matrix, H(t,u)\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R}) and H_u(t,u) denotes its gradient with respect to the u variable. We say that a solution u(t) of (1.1) is homoclinic (to 0) if u(t)\in C^2(\mathbb{R},{\mathbb{R}}^N) such that u(t)\to 0 and \dot{u}(t)\to 0 as |t|\to \mathrm{\infty }. If u(t)\not\equiv 0, then u(t) is called a nontrivial homoclinic solution.

If M=0 (zero matrix), then (1.1) reduces to the following second-order Hamiltonian system:

This is a classical equation which can describe many mechanic systems such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (1.2) have been studied by many authors via variational methods; see [1–18] and the references therein.

The periodic assumptions are very important in the study of homoclinic orbits for (1.2) since periodicity is used to control the lack of compactness due to the fact that (1.2) is set on all ℝ. However, non-periodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka [11] introduced a type of coercive condition on the matrix L(t),

and obtained the existence of a homoclinic orbit for non-periodic (1.2) under the Ambrosetti-Rabinowitz (AR) superquadratic condition:

where \mu >2 is a constant, (\cdot ,\cdot ) denotes the standard inner product in {\mathbb{R}}^N and the associated norm is denoted by |\cdot |.

We should mention that in the case where M\ne 0, i.e., the damped vibration system (1.1), only a few authors have studied homoclinic orbits of (1.1); see [19–23]. Zhu [23] considered the periodic case of (1.1) (i.e., L(t) and H(t,u) are T-periodic in t with T>0) and obtained the existence of nontrivial homoclinic solutions of (1.1). The authors [19–22] considered the non-periodic case of (1.1): Zhang and Yuan [22] obtained the existence of at least one homoclinic orbit for (1.1) when H satisfies the subquadratic condition at infinity by using a standard minimizing argument; by a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [21] obtained the existence and multiplicity of homoclinic orbits for (1.1) when H satisfies the local (AR) superquadratic growth condition:

where \mu >2 and r>0 are two constants. Notice that the authors [21, 22] all used condition (1.3). Recently, the author in [19, 20] obtained infinitely many homoclinic orbits for (1.1) when H satisfies the subquadratic[19] and asymptotically quadratic[20] condition at infinity by the following weaker conditions than (1.3):

(L₁) There is a constant \beta >1 such that

(L₂) There is a constant {\gamma }_0\ge 0 such that

which were firstly used in [15]. It is not hard to check that the matrix-valued function L(t):=(t^4{sin}^{2}t+1)I_N satisfies (L₁) and (L₂), but does not satisfy (1.3).

We define an operator \mathrm{\Gamma }:H^1(\mathbb{R},{\mathbb{R}}^N)\to H^1(\mathbb{R},{\mathbb{R}}^N) by

Since M is an antisymmetric N\times N constant matrix, Γ is self-adjoint on H^1(\mathbb{R},{\mathbb{R}}^N). Let χ denote the self-adjoint extension of the operator -\frac{d^2}{d{t}^2}+L(t)+\mathrm{\Gamma }. We are interested in the indefinite case:

(J₁) .

To state our main result, we still need the following assumptions:

(H₁) |\mathrm{\nabla }H(t,u)|\le c(1+{|u|}^{p-1}) for some c>0 and p>2, \mathrm{\forall }t\in \mathbb{R} and u\in {\mathbb{R}}^N.

(H₂) H(t,u)\ge \frac{1}{2}a{|u|}^2, \mathrm{\forall }t\in \mathbb{R} and u\in {\mathbb{R}}^N.

(H₃) For some \delta >0 and \gamma \in (0,b),

(H₄) \frac{H(t,u)}{{|u|}^2}\to +\mathrm{\infty } as |u|\to +\mathrm{\infty } and there exists W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}) such that

(H₅) For all t\in \mathbb{R} and u,z\in {\mathbb{R}}^N, there holds

Our main results read as follows.

Theorem 1.1If (L₁)-(L₂), (J₁) and (H₁)-(H₅) hold, then (1.1) has at least one nontrivial homoclinic orbit.

Theorem 1.2Let ℳ be the collection of solutions of (1.1), then there is a solution that minimizes the energy functional

over ℳ, where the spaceEis defined in Section  2. In addition, if

uniformly int, then there is a nontrivial homoclinic orbit that minimizes the energy functional over\mathcal{M}\setminus \{0\}, i.e., a ground state homoclinic orbit.

Remark 1.1 Although the authors [21] have studied (1.1) with superquadratic nonlinearities, our superquadratic condition (H₄) is weaker than (1.4) in [21]. Moreover, we study the ground state homoclinic orbit of (1.1). To the best of our knowledge, there has been no result published concerning the ground state homoclinic orbit of (1.1).

Example 1.1

1. (1)

   H(t,u)={|u|}^p,

2. (2)

   H(t,u)=g(t)({|u|}^p+(p-2){|u|}^{p-\epsilon }{sin}^2(\frac{{|u|}^{\epsilon }}{\epsilon })),

where p>2, g(t)>0 is continuous and . It is easy to check that the above two functions satisfy assumptions (H₁)-(H₅) if we take 0\le W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}), where W_1(t) is the function in (H₄)-(H₅).

The rest of the present paper is organized as follows. In Section 2, we establish the variational framework associated with (1.1), and we also give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

In the following, we use {\parallel \cdot \parallel }_{L^p} to denote the norm of L^p(\mathbb{R},{\mathbb{R}}^N) for any p\in [1,\mathrm{\infty }]. Let W:=H^1(\mathbb{R},{\mathbb{R}}^N) be a Hilbert space with the inner product and norm given respectively by

It is well known that W is continuously embedded in L^p(\mathbb{R},{\mathbb{R}}^N) for p\in [2,\mathrm{\infty }). We define an operator \mathrm{\Gamma }:W\to W by

Since M is an antisymmetric N\times N constant matrix, Γ is self-adjoint on W. Moreover, we denote by χ the self-adjoint extension of the operator -\frac{d^2}{d{t}^2}+L(t)+\mathrm{\Gamma } with the domain \mathcal{D}(\chi )\subset L^2(\mathbb{R},{\mathbb{R}}^N).

Let E:=\mathcal{D}({|\chi |}^{1/2}), the domain of {|\chi |}^{1/2}. We define respectively on E the inner product and the norm

where {(\cdot ,\cdot )}_2 denotes the inner product in L^2(\mathbb{R},{\mathbb{R}}^N).

By a similar proof of Lemma 3.1 in [15], we can prove that if conditions (L₁) and (L₂) hold, then

Therefore, it is easy to prove that the spectrum \sigma (\chi ) has a sequence of eigenvalues (counted with their multiplicities)

and the corresponding system of eigenfunctions \{e_k:k\in \mathbb{N}\} (\chi e_k={\lambda }_k{e}_k) forms an orthogonal basis in L^2(\mathbb{R},{\mathbb{R}}^N).

By (J₁), we may let

Then one has the orthogonal decomposition

with respect to the inner product {〈\cdot ,\cdot 〉}_E. Now, we introduce respectively on E the following new inner product and norm:

where u,v\in E=E^{-}\oplus E^{+} with u=u^{-}+u^{+} and v=v^{-}+v^{+}. Clearly, the norms \parallel \cdot \parallel and {\parallel \cdot \parallel }_E are equivalent (see [4]), and the decomposition E=E^{-}\oplus E^{+} is also orthogonal with respect to both inner products 〈\cdot ,\cdot 〉 and {(\cdot ,\cdot )}_2. Hence, by (J₁), E with equivalent norms, besides, we have

and

where a and b are defined in (J₁).

For problem (1.1), we consider the following functional:

Then I can be rewritten as

Let \mathrm{\Psi }(u):={\int }_{\mathbb{R}}H(t,u)dt. In view of the assumptions of H, we know I,\mathrm{\Psi }\in C^1(E,\mathbb{R}) and the derivatives are given by

for any u,v\in E=E^{-}\oplus E^{+} with u=u^{-}+u^{+} and v=v^{-}+v^{+}. By the discussion of [24], the (weak) solutions of system (1.1) are the critical points of the C^1 functional I:E\to \mathbb{R}. Moreover, it is easy to verify that if u\not\equiv 0 is a solution of (1.1), then u(t)\to 0 and \dot{u}(t)\to 0 as |t|\to \mathrm{\infty } (see Lemma 3.1 in [25]).

The following abstract critical point theorem plays an important role in proving our main result. Let E be a Hilbert space with the norm \parallel \cdot \parallel and have an orthogonal decomposition E=N\oplus N^{\mathrm{\perp }}, N\subset E is a closed and separable subspace. There exists a norm {|v|}_{\omega } satisfying {|v|}_{\omega }\le \parallel v\parallel for all v\in N and inducing a topology equivalent to the weak topology of N on a bounded subset of N. For u=v+w\in E=N\oplus N^{\mathrm{\perp }} with v\in N, w\in N^{\mathrm{\perp }}, we define {|u|}_{\omega }^2={|v|}_{\omega }^2+{\parallel w\parallel }^2. Particularly, if u_n=v_n+w_n is \parallel \cdot \parallel-bounded and u_n\stackrel{{|\cdot |}_{\omega }}{\to }u, then v_n\rightharpoonup v weakly in N, w_n\to w strongly in N^{\mathrm{\perp }}, u_n\rightharpoonup v+w weakly in E (cf.[26]).

Let E:=E^{-}\oplus E^{+}, z_0\in E^{+} with \parallel z_0\parallel =1. Let N:=E^{-}\oplus \mathbb{R}{z}_0 and E_1^{+}:=N^{\mathrm{\perp }}={(E^{-}\oplus \mathbb{R}{z}_0)}^{\mathrm{\perp }}. For R>0, let

with p_0=s_0{z}_0\in Q, s_0>0. We define

For I\in C^1(E,\mathbb{R}), define

where E_{\mathrm{fin}} denotes various finite-dimensional subspaces of E, \mathrm{\Gamma }\ne 0 since \mathit{id}\in \mathrm{\Gamma }.

The variant weak linking theorem is as follows.

Lemma 2.1 ([26])

The family ofC^1-functionals\{I_{\lambda }\}has the form

where{\lambda }_0>1. Assume that

1. (a)

   K(u)\ge 0, \mathrm{\forall }u\in E, I_1=I;

2. (b)

   |J(u)|+K(u)\to \mathrm{\infty }as\parallel u\parallel \to \mathrm{\infty };

3. (c)

   I_{\lambda }is{|\cdot |}_{\omega }-upper semicontinuous, I_{\lambda }^{\prime }is weakly sequentially continuous onE. Moreover, I_{\lambda }maps bounded sets to bounded sets;

4. (d)

   , \mathrm{\forall }\lambda \in [1,{\lambda }_0].

Then, for almost all\lambda \in [1,{\lambda }_0], there exists a sequence\{u_n\}such that

wherec_{\lambda }:={inf}_{h\in \mathrm{\Gamma }}{sup}_{u\in \overline{Q}}{I}_{\lambda }(h(t,u))\in [{inf}_D{I}_{\lambda },{sup}_{\overline{Q}}{I}_{\lambda }].

In order to apply Lemma 2.1, we shall prove a few lemmas. We pick {\lambda }_0 such that . For 1\le \lambda \le {\lambda }_0, we consider

It is easy to see that I_{\lambda } satisfies condition (a) in Lemma 2.1. To see (c), if u_n\stackrel{{|\cdot |}_{\omega }}{\to }u and I_{\lambda }(u_n)\ge c, then u_n^{+}\to u^{+} and u_n^{-}\rightharpoonup u^{-} in E, u_n\to u a.e. on ℝ, going to a subsequence if necessary. It follows from the weak lower semicontinuity of the norm, Fatou’s lemma and the fact H(t,u)+W_1(t)\ge 0 for all t\in \mathbb{R} and u\in {\mathbb{R}}^N by (1.5) in (H₄) that

Thus we get I_{\lambda }(u)\ge c. It implies that I_{\lambda } is {|\cdot |}_{\omega }-upper semicontinuous. I_{\lambda }^{\prime } is weakly sequentially continuous on E due to [27].

Lemma 2.2Under assumptions of Theorem  1.1, then

Proof By the definition of I(u) and (H₄), we have

which is due to W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}). □

Therefore, Lemma 2.2 implies that condition (b) holds. To continue the discussion, we still need to verify condition (d), that is, the following two lemmas.

Lemma 2.3Under assumptions of Theorem  1.1, there are two positive constantsϵ,\rho >0such that

Proof By (H₁), (H₃), (2.4) and the Sobolev embedding theorem, for all u\in E^{+},

where C is a positive constant. It implies the conclusion if we take \parallel u\parallel sufficiently small. □

Lemma 2.4Under assumptions of Theorem  1.1, then there is anR>0such that

whereQ_R:=\{u:=v+s{z}_0:s\ge 0,v\in E^{-},z_0\in E^{+}\mathit{\text{ with }}\parallel z_0\parallel =1,\parallel u\parallel \le R\}.

Proof Suppose by contradiction that there exist R_n\to \mathrm{\infty }, {\lambda }_n\in [1,{\lambda }_0] and u_n=v_n+s_n{z}_0\in \partial Q_R such that I_{{\lambda }_n}(u_n)>0. If s_n=0, then by (H₂) and (2.3), we have

Therefore, s_n\ne 0 and {\parallel u_n\parallel }^2={\parallel v_n\parallel }^2+s_n^2=R_n^2. Let {\tilde{u}}_n=\frac{u_n}{\parallel u_n\parallel }={\tilde{s}}_n{z}_0+{\tilde{v}}_n, then

It follows from I_{{\lambda }_n}(u_n)>0 and the definition of I that

There are renamed subsequences such that {\tilde{s}}_n\to \tilde{s}, {\lambda }_n\to \lambda, and there is a renamed subsequence such that {\tilde{u}}_n=\frac{u_n}{\parallel u_n\parallel }={\tilde{s}}_n{z}_0+{\tilde{v}}_n\rightharpoonup \tilde{u} in E and {\tilde{u}}_n\to \tilde{u} a.e. on ℝ.

We claim that

Case 1. If \tilde{u}\not\equiv 0. Let {\mathrm{\Omega }}_0 be the subset of ℝ where \tilde{u}\ne 0, then for all t\in {\mathrm{\Omega }}_0 we have |u_n|=|{\tilde{u}}_n|\cdot \parallel u_n\parallel \to \mathrm{\infty }. It follows from (H₄) and W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}) that

Case 2. If \tilde{u}\equiv 0, then by (H₄) and W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}), we have

Therefore, Cases 1 and 2 imply that (2.6) holds. Therefore, by (2.5), (2.6) and the facts {\tilde{s}}_n\to \tilde{s}, {\lambda }_n\to \lambda, we have

that is, {\tilde{s}}^2\ge \frac{1}{1+\lambda }\ge \frac{1}{1+{\lambda }_0}>0. Thus, \tilde{u}\ne 0. It follows from (H₄) that

which contradicts (2.5). The proof is finished. □

Therefore, Lemmas 2.3 and 2.4 imply that condition (d) of Lemma 2.1 holds. Applying Lemma 2.1, we soon obtain the following fact.

Lemma 2.5Under assumptions of Theorem  1.1, for almost all\lambda \in [1,{\lambda }_0], there exists a sequence\{u_n\}such that

where the definition ofc_{\lambda }is given in Lemma  2.1.

Lemma 2.6Under assumptions of Theorem  1.1, for almost all\lambda \in [1,{\lambda }_0], there exists au_{\lambda }\in Esuch that

Proof Let \{u_n\} be the sequence obtained in Lemma 2.5. Since \{u_n\} is bounded, we can assume u_n\rightharpoonup u_{\lambda } in E and u_n\to u_{\lambda } a.e. on ℝ. By (H₁), (H₃), (2.1) and Theorem A.4 in [27], we have

and

By Lemma 2.5 and the fact that I_{\lambda }^{\prime } is weakly sequentially continuous, we have

That is, I_{\lambda }^{\prime }(u_{\lambda })=0. By Lemma 2.5, we have

It follows from (2.7), (2.8) and the fact I_{\lambda }^{\prime }(u_{\lambda })=0 that

The proof is finished. □

Applying Lemma 2.6, we soon obtain the following fact.

Lemma 2.7Under assumptions of Theorem  1.1, for every\lambda \in [1,{\lambda }_0], there are sequences\{u_n\}\subset Eand{\lambda }_n\in [1,{\lambda }_0]with{\lambda }_n\to \lambdasuch that

Lemma 2.8Under assumptions of Theorem  1.1, then

whereu\in E, w\in E^{+}, 0\le r\le 1and the constantC:={\int }_{\mathbb{R}}|W_1(t)|dtdoes not depend onu, w, r.

Proof This follows from (H₅) if we take u=u and z=rw-u. □

Lemma 2.9The sequences given in Lemma  2.7 are bounded.

Proof Write u_n=u_n^{+}+u_n^{-}, where u_n^{\pm }\in E^{\pm }. Suppose that

Let v_n:=\frac{u_n}{\parallel u_n\parallel }, then v_n^{+}=\frac{u_n^{+}}{\parallel u_n\parallel }, v_n^{-}=\frac{u_n^{-}}{\parallel u_n\parallel }, {\parallel v_n\parallel }^2={\parallel v_n^{+}\parallel }^2+{\parallel v_n^{-}\parallel }^2=1 and \parallel v_n^{+}\parallel \le 1. Thus v_n^{+}\rightharpoonup v^{+} in E and v_n^{+}\to v^{+} a.e. on ℝ, after passing to a subsequence.

Case 1. If v^{+}\not\equiv 0. Let {\mathrm{\Omega }}_1 be the subset of ℝ where v^{+}\ne 0. Then v\ne 0 and |u_n|=|v_n|\cdot \parallel u_n\parallel \to \mathrm{\infty } on {\mathrm{\Omega }}_1. It follows from (H₄) and W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}) that

which together with Lemmas 2.3 and 2.7 and v_n^{\pm }\to v^{\pm } in L^q(\mathbb{R},{\mathbb{R}}^N) for all 1\le q\le \mathrm{\infty } (by (2.1)) implies that

It is a contradiction.

Case 2. If v^{+}\equiv 0. We claim that there is a constant C independent of u_n and {\lambda }_n such that

Since

it follows from the definition of I that

Take \phi :=(r^2+1)u_n^{-}-(r^2-1)u_n^{+}=(r^2+1)u_n-2r^2{u}_n^{+} in (2.10), then it follows from Lemma 2.8 that

Thus (2.9) holds.

Let C_0>0 be a fixed constant and take

Therefore, (2.9) implies that

It follows from v_n^{+}=\frac{u_n^{+}}{\parallel u_n\parallel } and Lemma 2.7 that

Note that Lemmas 2.3 and 2.7 and (H₄) imply that

It follows from the fact \frac{{\int }_{\mathbb{R}}{W}_1(t)dt}{{\parallel u_n\parallel }^2}\to 0 as n\to \mathrm{\infty } due to W_1(t)\in L^1(\mathbb{R},{\mathbb{R}}^{+}) that

for all sufficiently large n. We take \epsilon =\frac{1}{4}, by (2.12) and {\parallel v_n\parallel }^2={\parallel v_n^{+}\parallel }^2+{\parallel v_n^{-}\parallel }^2=1, we have

for all sufficiently large n. By (H₁) and (H₃), we have

For all sufficiently large n, by (2.13) and (2.14), it follows from {\lambda }_n\to \lambda and v_n^{+}\to v^{+}\equiv 0 in L^q(\mathbb{R},{\mathbb{R}}^N) for all 1\le q\le \mathrm{\infty } (by (2.1)) that

This implies that I_{{\lambda }_n}(C_0{v}_n^{+})\to \mathrm{\infty } as C_0\to \mathrm{\infty }, contrary to (2.11).

Therefore, \{u_n\} are bounded. The proof is finished. □

Proof of Theorem 1.1 From Lemma 2.7, there are sequences and \{u_n\}\subset E such that I_{{\lambda }_n}^{\prime }(u_n)=0 and I_{{\lambda }_n}(u_n)=c_{{\lambda }_n}. By Lemma 2.9, we know that \{u_n\} is bounded in E. Thus we can assume u_n\rightharpoonup u in E, u_n\to u a.e. on ℝ. Therefore,

Hence, in the limit,

Thus I^{\prime }(u)=0. Note that

Similar to (2.7) and (2.8), we know