Ground state homoclinic orbits of damped vibration problems

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.

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.

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

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):

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).

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) $a:=sup(\sigma (\chi )\cap (-\mathrm{\infty },0))<0<b:=inf(\sigma (\chi )\cap (0,\mathrm{\infty }))$.

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 $u\in {\mathbb{R}}^N$.

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.

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).

where $p>2$, $g(t)>0$ is continuous and $0<\epsilon <p-2$. 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.

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 $\mathcal{D}(\chi )\subset L^2(\mathbb{R},{\mathbb{R}}^N)$.

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

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

where a and b are defined in (J₁).

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]).

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])

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

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].

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.

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

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

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.

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.

where the definition of$c_{\lambda }$is given in Lemma  2.1.

The proof is finished. □

Applying Lemma 2.6, we soon obtain the following fact.

where$u\in E$, $w\in E^{+}$, $0\le r\le 1$and the constant$C:={\int }_{\mathbb{R}}|W_1(t)|dt$does 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.

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.

It is a contradiction.

Thus (2.9) holds.