Low energy small solutions of systems of Kirchhoff-type equations on R N

Abstract

Low energy-induced vibration has been ignored, which, however, might lead to damage of a structure. The main purpose of this paper is to study the low energy small solution sequence of systems of Kirchhoff-type equations by the variational approaches. Our result shows that the low energy vibration can be controlled by setting the parameters and this can be used for optimization of structures under small perturbations, e.g. the bridge vibration under winds.

Introduction and main result

Vibration is the mechanical oscillation of a particle, member, or a body from its position of equilibrium. The vibrations sometimes may be very weak for identification and sometimes may be large and devastating such as earthquakes, winds, and tsunamis.^1,2 In vibration study, there are two types of systems – discrete and continuous. Discrete systems are described mathematically by the variables that depend only on time. On the other hand, continuous systems are described by variables that depend on time and space. And therefore, the equations of motion of discrete systems are described by ordinary differential equations, whereas the equations of motion for continuous systems are governed by partial differential equations. There have been many interesting studies in recent years of the nonlinear behaviour of strings in vibration, see Stulov and Kartofelev, ³ Issanchou et al., ⁴ Horgan and Chan, ⁵ Peradze, ⁶ and therein. The Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations (see Peradze ⁶ ). It was proposed by Kirchhoff ⁷ as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. In recent years, many kinds of Kirchhoff-type equations are derived and studied by various methods, for example, in the literature.^6,8–13,15

In this paper, we consider the following systems of Kirchhoff-type equations

$$\{\begin{array}{l}-\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\text{d}x\right)\Delta u+V(x)u=F_u(x,u,v),x\in {\mathbb{R}}^N,\hfill \\ -\left(c+d{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v{|}^2\text{d}x\right)\Delta v+V(x)v=F_v(x,u,v),x\in {\mathbb{R}}^N,\hfill \\ u(x)\to 0\mathrm{and}v(x)\to 0\mathrm{as}\left|x\right|\to \infty \hfill \end{array}$$

(1)

where constants $a,c>0,b,d\ge 0$ , $F\in C^1\left({\mathbb{R}}^N\times {\mathbb{R}}^2,\mathbb{R}\right)$ , $F_u=\frac{\partial F}{\partial u},F_v=\frac{\partial F}{\partial v}$ and $V\in C\left({\mathbb{R}}^N,\mathbb{R}\right)$ .

Assume that $I\left(u,v\right)$ is the energy functional of system (1). A weak solution sequence $\left\{\left(u_n,v_n\right)\right\}$ of system (1) is said to be a low energy solution sequence if the energy $I(u_n,v_n)\to 0$ as $n\to \infty$ . A low energy solution sequence $\left\{\left(u_n,v_n\right)\right\}$ is a small solution sequence if $(u_n,v_n)\to (0,0)$ as $n\to \infty$ .

Cheng et al. ⁹ studied a $\left(p,q\right)$ -Kirchhoff type system. Recently, Wu ¹⁴ obtained a sequence of high energy solutions of system (1). After presenting a new proof technique, Zhou et al. ¹⁶ improved the results obtained in Wu. ¹⁴

Motivated largely by the work of Zhou et al., ¹⁶ we are going to study the low energy solutions of system (1) by the variational approaches in this paper.

In order to state our main result, we need the following assumptions.

$\left(V\right)$ $V\in C\left({\mathbb{R}}^N,\mathbb{R}\right)$ satisfies $\inf \hspace{0.17em}V(x)\ge a_1>0$ and for each $M>0$ , $meas\{x\in {\mathbb{R}}^N:V(x)\le M\}<+\infty$ , where $a_1$ is a constant and $meas$ denote the Lebesgue measure in ${\mathbb{R}}^N$ .

$(\hspace{0.17em}{f}_1)$ There exist $p\in (1,2)$ , $q\in [2,2^{*})$ , $C>0$ and a positive continuous function $\beta \in L^{\frac{2}{2-p}}\left({\mathbb{R}}^N\right)\cap L^{\infty }\left({\mathbb{R}}^N\right)$ such that

$$\left|F_u(x,u,v)\right|+\left|F_v(x,u,v)\right|\le \beta (x){\left|(u,v)\right|}^{p-1}+C{\left|(u,v)\right|}^{q-1}$$

for all $(x,u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^2$ $(\hspace{0.17em}{f}_2)$ There exist $\delta >0$ , $\sigma \in (1,2)$ and a positive continuous function $\zeta \in L^{\frac{2}{2-\sigma }}({\mathbb{R}}^N)$ such that

$$u{F}_u(x,u,v)+v{F}_v(x,u,v)\ge \zeta (x){\left|(u,v)\right|}^{\sigma }$$

for $\left|(u,v)\right|\le \delta$ and $x\in {\mathbb{R}}^N$ $(\hspace{0.17em}{f}_3)$ $F\left(x,-u,-v\right)=F\left(x,u,v\right)$ for all $(x,u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^2$

Remark: Here, we need to point out that only the conditions of potential $\left(V\right)$ and $(\hspace{0.17em}{f}_3)$ , which is a basic condition for finding infinitely many solutions, are the same with Zhou et al. ¹⁶ Moreover, Zhou et al. ¹⁶ mainly used the $Z_2$ version of the mountain pass theorem to prove the results, while our proof depends on a cut-off technique and the below version of mountain pass theorem.

Our main result is the following theorem.

Theorem 1. Assume that conditions $\left(V\right)$ and $(\hspace{0.17em}{f}_1)$ – $(\hspace{0.17em}{f}_3)$ hold. Then the system (1) possesses infinitely many distinct pairs of low energy small solutions.

Preliminaries

Let

$$H^1({\mathbb{R}}^N)=\left\{u\in L^2({\mathbb{R}}^N):\nabla u\in L^2({\mathbb{R}}^N)\right\}$$

with the norm

$${\Vert u\Vert }_{H^1}={\left({\displaystyle {\int }_{{\mathbb{R}}^N}\left({\left|\nabla u\right|}^2+u^2\right)\mathrm{d}x}\right)}^{{\hspace{0.17em}}^{\frac{1}{2}}}$$

Let

$$X=\left\{u\in H^1({\mathbb{R}}^N):{\displaystyle {\int }_{{\mathbb{R}}^N}\left({\left|\nabla u\right|}^2+V(x)u^2\right)\mathrm{d}x<+}\infty \right\}$$

with the inner product and the norm

$${〈u,v〉}_X={\displaystyle {\int }_{{\mathbb{R}}^N}\left(\nabla u\cdot \nabla v+V(x)uv\right)\mathrm{d}x,}{\Vert u\Vert }_X={〈u,u〉}_X^{\frac{1}{2}}$$

Then $X$ is a Hilbert space. Moreover, it is well known that the embedding $X$ $\subset L^s\left({\mathrm{ℝ}}^N\right)$ , $2\le s<2^{*}$ is compact under the assumption $\left(V\right)$ .

For any $(u,v),(\phi ,\psi )\in X\times X$ , define an inner product $〈\cdot ,\cdot 〉$ and a norm $\Vert ·\Vert$ on $X\times X$ by

$$〈(u,v),(\phi ,\psi )〉=〈u,\phi 〉+〈v,\psi 〉$$

and

$${\Vert \left(u,v\right)\Vert }^2=〈(u,v),(u,v)〉={\Vert u\Vert }_X^2+{\Vert v\Vert }_X^2$$

Then $E=X\times X$ is a Hilbert space and by the semi-inverse method^17–19 we see that a weak solution of the system (1) is a critical point on $E$ of the following functional $I$

$$\begin{array}{l}I(u,v)=\frac{a}{2}{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\mathrm{d}x+\frac{b}{4}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\mathrm{d}x\right)}^2+\frac{1}{2}{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)u^2\mathrm{d}x\\ +\frac{c}{2}{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v{|}^2\mathrm{d}x+\frac{d}{4}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v{|}^2\mathrm{d}x\right)}^2+\frac{1}{2}{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)v^2\mathrm{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}F}(x,u,v)\mathrm{d}x\end{array}$$

By a standard argument, we have the following Lemma.

Lemma 1. If assumptions $\left(V\right)$ and $\left(f_1\right)$ hold, then $I\in C^1(E,\mathbb{R})$

$$\begin{array}{l}〈I^{\prime }(u,v),(\phi ,\psi )〉=\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\text{d}x\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla }u\nabla \phi \text{d}x+{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)u\phi \text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_u}(x,u,v)\phi \text{d}x\\ +\left(c+d{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v{|}^2\text{d}x\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla }v\nabla \psi \text{d}x+{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)v\psi \text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_v}(x,u,v)\psi \text{d}x\end{array}$$

and $\Psi \mathrm{\prime }:E\to E^{\ast }$ is compact, where $\Psi (u,v)={\int }_{{\mathbb{R}}^N}F(x,u,v)\text{d}x$ .

Recall that a sequence $\left\{\left(u_n,v_n\right)\right\}$ is called a Palais–Smale (PS) of $I$ if $I\left(u_n,v_n\right)$ is bounded and $I\mathrm{\prime }(u_n,v_n)\to 0$ . The functional $I$ satisfies the PS condition if every PS sequence has a convergent sequence. The next lemma gives the convergent properties of a bounded PS sequence.

Lemma 2. If $\left(f_1\right)$ holds, then every bounded PS sequence $\{(u_n,v_n)\}\subset E$ of $I$ has a convergent subsequence.Proof. Since $\{(u_n,v_n)\}\subset E$ is bounded and the embedding $X$ $\subset L^s\left({\mathrm{ℝ}}^N\right)$ is compact for each $s\in [2,2^{\ast })$ , passing a subsequence, we can assume that $u_n\rightharpoonup u,v_n\rightharpoonup v$ in $X$ and $u_n\to u,v_n\to vin{L}^s({\mathbb{R}}^N)$ for $s\in [2,2^{\ast })$ . Clearly

$$\begin{array}{l}\min \{a,1\}{\Vert u_n-u\Vert }_X^2\le 〈I\mathrm{\prime }\left(u_n,v_n\right)-I\mathrm{\prime }\left(u,v\right),\left(u_n-u,0\right)〉\\ +b\left({\displaystyle {\int }_{{\mathbb{R}}^N}|\nabla u_n{|}^2\text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}|\nabla u_n{|}^2\text{d}x}}\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla u\cdot \nabla (u_n-u)\text{d}x}\\ +{\displaystyle {\int }_{{\mathbb{R}}^N}\left[F_u\left(x,u_n,v_n\right)-F_u\left(x,u,v\right)\right]\left(u_n-u\right)\text{d}x}\end{array}$$

The boundedness of $\left\{u_n\right\}$ in $X$ implies

$$b\left({\displaystyle {\int }_{{\mathbb{R}}^N}|\nabla u{|}^2\text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}|\nabla u_n{|}^2}\text{d}x}\right){\displaystyle {\int }_{{\mathbb{R}}^N}{\nabla }_u}\cdot \nabla (u_n-u)\text{d}x\to 0$$

as $n\to +\infty .$ Moreover, by $\left(f_1\right)$ and Hölder inequality, we have

$$\begin{array}{l}\left|{\displaystyle {\int }_{{\mathbb{R}}^N}\left[F_u\left(x,u_n,v_n\right)-Fu\left(x,u,v\right)\right]}\left(u_n-u\right)\text{d}x\right|\\ \le {\displaystyle {\int }_{{\mathbb{R}}^N}\left[\beta (x){\left|(u_n,v_n)\right|}^{p-1}+C{\left|\left(u_n,v_n\right)\right|}^{q-1}+\beta (x){\left|\left(u,v\right)\right|}^{p-1}+C{\left|\left(u,v\right)\right|}^{q-1}\right]}(u_n-u)\text{d}x\\ \le \left[{\left({\displaystyle {\int }_{{\mathbb{R}}^N}\beta }(x)|(u_n,v_n){|}^p\text{d}x\right)}^{\frac{p-1}{p}}+{\left({\displaystyle {\int }_{{\mathbb{R}}^N}\beta }(x)|(u,v){|}^p\text{d}x\right)}^{\frac{p-1}{p}}\right]{\left({\displaystyle {\int }_{{\mathbb{R}}^N}\beta }(x)|u_n-u{|}^p\text{d}x\right)}^{\frac{1}{p}}\\ +C\left[{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}(u_n,v_n){|}^q\text{d}x\right)}^{\frac{q-1}{q}}+{({\displaystyle {\int }_{{\mathbb{R}}^N}|}(u,v){|}^q\text{d}x)}^{\frac{q-1}{q}}\right]{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}{u}_n-u{|}^q\text{d}x\right)}^{\frac{1}{q}}\\ \le \left[{\Vert \beta \Vert }_{\frac{2}{2-p}}^{\frac{p-1}{p}}{\Vert (u_n,v_n)\Vert }_2^{p-1}+{\Vert \beta \Vert }_{\frac{2}{2-p}}^{\frac{p-1}{p}}{\Vert (u,v)\Vert }_2^{p-1}\right]{\Vert \beta \Vert }_{\frac{2}{2-p}}^{\frac{1}{p}}{\Vert (u_n-u)\Vert }_2\\ +C\left[{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}(u_n,v_n){|}^q\text{d}x\right)}^{\frac{q-1}{q}}+{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}(u,v){|}^q\text{d}x\right)}^{\frac{q-1}{q}}\right]{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}{u}_n-u{|}^q\text{d}x\right)}^{\frac{1}{q}}\\ \le C_1{\Vert (u_n-u)\Vert }_2+C_2{\Vert (u_n-u)\Vert }_q\end{array}$$

where ${\Vert \left(u_n,v_n\right)\Vert }_2={\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}(u_n,v_n){|}^2\text{d}x\right)}^{\frac{1}{2}}$ . Consequently, by $I\mathrm{\prime }\left(u_n,v_n\right)\to 0$ and $u_n\rightharpoonup u$ in $X$ , we know that ${\Vert u_n-u\Vert }_X\to 0$ . Using the same method, we can prove ${\Vert v_n-v\Vert }_X\to 0$ . Hence $\left(u_n,v_n\right)\to \left(u,v\right)$ in $E$ . This completes the proof.

Lemma 3. If $(u,v)\in E$ is a critical point of $I$ , then $u(x)\to 0$ and $v(x)\to 0$ as $\left|x\right|\to \infty$ .

Proof. The conclusion can be obtained by De Giorgi’s iteration (see Liu et al. ²⁰ (^pp898–899)).

Existence of low energy small solutions

We consider the existence of low energy small solutions of system (1) in this section. Our proof of Theorem 1 depends on a cut-off technique and the below version of mountain pass theorem due to Kajikiya. ²¹

Theorem 2. Let $E$ be an infinite dimensional Banach space and $I\in C^1\left(E,\mathbb{R}\right)$ satisfies $\left(A_1\right)$ and $\left(A_2\right)$ below.

$\left(A_1\right)$ $I$ is even, bounded from below, $I\left(0\right)=0$ and $I$ satisfies the PS condition.

$\left(A_2\right)$ For each $k\in \mathbb{N}$ , there exists an $A_k\in {\Gamma }_k$ such that $\underset{u\in A_k}{\sup }I(u)<0$ , where ${\Gamma }_k=\left\{A\subset E:A\text{is a closed symmetric set such that}0\notin A\text{and the genus}\gamma (A)\ge k\right\}$

Then either $\left(i\right)$ or $\left(ii\right)$ below holds.

$\left(i\right)$ There exists a sequence $\left\{u_n\right\}$ such that $I\mathrm{\prime }\left(u_n\right)=0$ , $I\left(u_n\right)<0$ and $u_n\to 0$ . $\left(ii\right)$ There exist two sequences $\left\{u_n\right\}$ and $\left\{v_n\right\}$ such that $I\mathrm{\prime }\left(u_n\right)=0$ , $I\left(u_n\right)=0$ , $u_n\ne 0$ , $u_n\to 0$ , $I\mathrm{\prime }\left(v_n\right)=0$ , $I\left(v_n\right)<0$ , and $I\left(v_n\right)\to 0$ and $\left\{v_n\right\}$ converges to a nonzero limit.

Choose $l\le \frac{\delta }{2}$ , where $\delta$ is the constant appearing in $\left(f_2\right)$ . We take an even function $h\in C^1(\mathbb{R},\mathbb{R})$ such that $0\le h(t)\le 1$ , $h\left(t\right)=1$ for $\left|t\right|\le l$ , $h\left(t\right)=0$ for $\left|t\right|\ge 2l$ , and $h\left(t\right)$ is decreasing in $\left[l,2l\right]$ . Consider the equations

$$\{\begin{array}{l}-\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\text{d}x\right)\Delta u+V(x)u=F_u(x,u,v)h\left(|(u,v)|\right),x\in {\mathbb{R}}^N,\hfill \\ -\left(c+d{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v{|}^2\text{d}x\right)\Delta v+V(x)v=F_v(x,u,v)h\left(|(u,v)|\right),x\in {\mathbb{R}}^N,\hfill \\ u(x)\to 0\mathrm{ and }v(x)\to 0\mathrm{as}\left|x\right|\to \infty \hfill \end{array}$$

(2)

We would like to prove that equation (2) has a sequence $\left\{\left(u_n,v_n\right)\right\}$ of weak solutions with $(u_n,v_n)\to (0,0)$ in $E$ and ${\Vert \left(u_n,v_n\right)\Vert }_{L^{\infty }}\le l$ for all $n$ , then the sequence $\left\{\left(u_n,v_n\right)\right\}$ will be the weak solutions sequence of equation (1).

Define a functional $I_h$ on $E$ by

$$I_h(u,v)=\frac{a}{2}{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\text{d}x+\frac{b}{4}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\text{d}x\right)}^2+\frac{1}{2}{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)u^2\text{d}x+\frac{c}{2}{\int }_{{\mathrm{ℝ}}^N}|\nabla v{|}^2\text{d}x+\frac{d}{4}{\left({\int }_{{\mathrm{ℝ}}^N}\right|\nabla v{|}^2\text{d}x)}^2+\frac{1}{2}{\int }_{{\mathrm{ℝ}}^N}V(x)v^2\text{d}x-{\int }_{{\mathrm{ℝ}}^N}{F}_h(x,u,v)\text{d}x$$

for all $(u,v)\in E$ , where

$$F_h(x,u,v)={\displaystyle {\int }_0^1\left[u{F}_u(x,tu,tv)h\left(\left|(tu,tv)\right|\right)+v{F}_v(x,tu,tv)h\left(\left|(tu,tv)\right|\right)\right]\text{d}t}$$

Since the functions $F_u\left(x,u,v\right)h\left(\left|\left(u,v\right)\right|\right)$ and $F_v\left(x,u,v\right)h\left(\left|\left(u,v\right)\right|\right)$ satisfy condition $\left(f_1\right)$ , Lemma 1 and 2 still hold as the functions $F_u\left(x,u,v\right)h\left(\left|\left(u,v\right)\right|\right)$ and $F_v\left(x,u,v\right)h\left(\left|\left(u,v\right)\right|\right)$ are used in place of the functions $F_u\left(x,u,v\right)$ and $F_v\left(x,u,v\right)$ , respectively. Hence $I_h\in C^1(E,\mathbb{R})$ and for every $(\phi ,\psi )\in E$ , there holds

$$\begin{array}{l}〈I_{h^{\mathrm{\prime }}}(u,v),(\phi ,\psi )〉\\ =\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u{|}^2\text{d}x\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla }u\nabla \phi \text{d}x+{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)u\phi \text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_u}(x,u,v)h(|(u,v)|)\phi \text{d}x\\ +\left(c+d{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v{|}^2\text{d}x\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla }v\nabla \psi \text{d}x+{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)v\psi \text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_v}(x,u,v)h(|(u,v)|)\psi \text{d}x\end{array}$$

and every bounded PS sequence $\left\{\left(u_n,v_n\right)\right\}\subset E$ of $I_h$ has a convergent subsequence.

We need the following Lemmas.

Lemma 4. Let $\tilde{E}$ be a finite dimensional subspace of $E$ . Then for any positive integer $k$ , there exists $n_k>k$ such that

$$\text{meas}\left\{x\in {\mathbb{R}}^N:\zeta (x)h(|(u,v)|)|(u,v){|}^{\sigma }\ge n_k^{\sigma }{\Vert (u,v)\Vert }_2^2\right\}\ge \frac{1}{n_k}$$

for any $(u,v)\in \tilde{E}\backslash \{(0,0)\}$ with ${\Vert \left(u,v\right)\Vert }_2<{\left(\frac{1}{n_k}\right)}^{\frac{\sigma +1}{2-\sigma }}$ , where ${\Vert \left(u,v\right)\Vert }_2^2={\displaystyle {\int }_{{\mathbb{R}}^N}|}(u,v){|}^2\text{d}x$ .

Proof. If the conclusion is false, then there exist $k_0>0$ and $(u_n,v_n)\in \tilde{E}\backslash \left\{(0,0)\right\}$ with ${\Vert \left(u_n,v_n\right)\Vert }_2<{\left(\frac{1}{n}\right)}^{\frac{\sigma +1}{2-\sigma }}$ such that

$$\text{meas}\left\{x\in {\mathbb{R}}^N:\zeta (x)h(|(u_n,v_n)|)|(u_n,v_n){|}^{\sigma }\ge n^{\sigma }{\Vert \left(u_n,v_n\right)\Vert }_2^2\right\}<\frac{1}{n}$$

for all positive integer $n>k_0$ . Take $w_n=\frac{|(u_n,v_n)|}{{\Vert \left(u_n,v_n\right)\Vert }_2}$ , then $w_n$ is well defined since $(u_n,v_n)\ne (0,0)$ . Note that $\tilde{E}$ is finite dimensional. There exists a finite dimensional subspace $\tilde{X}$ of $X$ such that $|(u_n,v_n)|\in \tilde{X}$ for every $n$ . Moreover, $\left\{w_n\right\}$ is bounded in $\tilde{X}$ since all norms are equivalent in a dimensional space. We can assume that, up to a subsequence, $w_n\to w$ in $X$ and hence $w_n\to w$ in $L^2({\mathbb{R}}^N)$ , $w_n(x)\to w(x)$ a.e. $x\in {\mathbb{R}}^N$ and $|(u_n,v_n)|\to 0$ a.e. $x\in {\mathbb{R}}^N$ .

Since ${\Vert \left(u_n,v_n\right)\Vert }_2<{\left(\frac{1}{n}\right)}^{\frac{\sigma +1}{2-\sigma }}$ and $\text{meas}\left\{x\in {\mathbb{R}}^N:\zeta (x)h\left(|(u_n,v_n)|\right)|w_n{|}^{\sigma }\ge n^{\sigma }{\Vert w_n\Vert }_2^{2-\sigma }\right\}<\frac{1}{n}$ for all $n>k_0$ , we obtain $\text{meas}\left\{x\in {\mathbb{R}}^N:\zeta (x)h\left(|(u_n,v_n)|\right)|w_n{|}^{\sigma }\ge \frac{1}{n}\right\}<\frac{1}{n}$ for all $n>k_0$ , which implies that

$\zeta (x)h\left(\left|\left(u_n,v_n\right)\right|\right){\left|w_n\right|}^{\sigma }$ converges to 0 in measure.

On the other hand, it is obvious that $\zeta (x)h\left(\left|\left(u_n,v_n\right)\right|\right){\left|w_n\right|}^{\sigma }$ converges to $\zeta (x)|w{|}^{\sigma }$ in measure. Since ${\Vert w\Vert }_2=1$ and $\zeta (x)$ is positive, $\zeta (x)|v{|}^{\sigma }\ne 0$ . This is a contradiction. The proof is completed.

Lemma 5. If $\left\{\left(u_n,v_n\right)\right\}$ is a sequence of critical points of $I_h$ with $(u_n,v_n)\to (0,0)$ in $E$ , then, up to a subsequence, ${\Vert \left(u_n,v_n\right)\Vert }_{L^{\infty }}\le l$ for all $n$ .

Proof. For $T>\frac{l}{2}$ and $(u,v)\in E$ , define $u^T=u$ for $\left|u\right|\le T$ and $u^T=(\text{sign}u)T$ for $\left|u\right|>T$ .

Noting that $(u_n,v_n)$ is a critical point of $I_h$ , we have

$$\begin{array}{l}0=\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla u_n{|}^2\text{d}x\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla }{u}_n\nabla \phi \text{d}x+{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)u_n\phi dx-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_u}(x,u_n,v_n)h(|(u_n,v_n)|)\phi \text{d}x\\ +\left(c+d{\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla v_n{|}^2\text{d}x\right){\displaystyle {\int }_{{\mathbb{R}}^N}\nabla }{v}_n\nabla \psi \text{d}x+{\displaystyle {\int }_{{\mathbb{R}}^N}V}(x)v_n\psi \text{d}x-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_v}(x,u_n,v_n)h\left(|(u_n,v_n)|\right)\psi \text{d}x\end{array}$$

(3)

for any $(\phi ,\psi )\in E$ . Define $w_n=\{\begin{array}{ll}{u}_n,\hfill & \left|u_n\right|\le \frac{l}{2\sqrt{2}}\hfill \\ 0,\hfill & \left|u_n\right|>\frac{l}{2\sqrt{2}}\hfill \end{array}$ and ${\chi }_n=\{\begin{array}{ll}0,\hfill & \left|u_n\right|\le \frac{l}{2\sqrt{2}}\hfill \\ u_n,\hfill & \left|u_n\right|>\frac{l}{2\sqrt{2}}\hfill \end{array}$ . For $r\ge 0$ , taking $(\phi ,\psi )=\left({\chi }_n{\left|{\chi }_n^T\right|}^{2r},0\right)$ as a test function in equation (3), we have

$$\begin{array}{l}{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_u}(x,u_n,v_n)h\left(\left|\left(u_n,v_n\right)\right|\right){\chi }_n|{\chi }_n^T{|}^{2r}\text{d}x\ge a\left({\displaystyle {\int }_{{\mathbb{R}}^N}|}\nabla {\chi }_n{|}^2|{\chi }_n^T{|}^{2r}\text{d}x+{\displaystyle {\int }_{\left|{\chi }_n\right|\le T}2}r|\nabla {\chi }_n{|}^2|{\chi }_n{|}^{2r}\text{d}x\right)\\ \ge \frac{2a}{r+2}{\displaystyle {\int }_{{\mathbb{R}}^N}\left({\left|\nabla {\chi }_n\right|}^2|{\chi }_n^T{|}^{2r}+(r^2+2r)|\nabla {\chi }_n^T{|}^2|{\chi }_n{|}^{2r}\right)\text{d}x}\\ =\frac{2a}{r+2}{\displaystyle {\int }_{{\mathbb{R}}^N}{\left|\nabla \left({\chi }_n|{\chi }_n^T{|}^r\right)\right|}^2}\text{d}x\\ \ge \frac{c}{r+2}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n{\left|{\chi }_n^T\right|}^r\right|}^{2^{*}}\text{d}x}\right)}^{\frac{2}{2^{*}}}\end{array}$$

By $\left(f_1\right)$ , there exists constants $c_0$ such that

$$\left|F_u\left(x,u,v\right)h\left(\left|\left(u,v\right)\right|\right)\right|\le c_0\left|u\right|$$

for $\left|u\right|\ge \frac{l}{2\sqrt{2}}$ and $\left|(u,v)\right|\le 2l$ , which implies

$${\displaystyle {\int }_{{\mathbb{R}}^N}{F}_u}\left(x,u_n,v_n\right)h\left(\left|\left(u_n,v_n\right)\right|\right){\chi }_n{\left|{\chi }_n^T\right|}^{2r}\text{d}x\le c_0{\displaystyle {\int }_{{\mathbb{R}}^N}\left|u_n\right|\left|{\chi }_n\right|{\left|{\chi }_n^T\right|}^{2r}}\text{d}x\le c_0{\displaystyle {\int }_{{\mathbb{R}}^N}{\left(\left|{\chi }_n\right|{\left|{\chi }_n^T\right|}^r\right)}^2}\text{d}x$$

Hence

$${\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n{\left|{\chi }_n^T\right|}^r\right|}^{2^{*}}\text{d}x}\right)}^{\frac{1}{2^{*}(r+1)}}\le {\left[c_0(r+1)\right]}^{\frac{1}{2(r+1)}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left(\left|{\chi }_n\right|{\left|{\chi }_n^T\right|}^r\right)}^2}\text{d}x\right)}^{\frac{1}{2(r+1)}}$$

Taking $T\to \infty$ , one has

$${\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n\right|}^{2^{*}(r+1)}\text{d}x}\right)}^{\frac{1}{2^{*}(r+1)}}\le {\left[c_0(r+1)\right]}^{\frac{1}{2(r+1)}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n\right|}^{2(r+1)}\text{d}x}\right)}^{\frac{1}{2(r+1)}}$$

Let $r_0=0$ and $2(r_{k+1}+1)=2^{*}(r_k+1)$ . Then

$$\begin{array}{l}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n\right|}^{2(r_{k+1}+1)}\text{d}x}\right)}^{\frac{1}{2(r_{k+1}+1)}}\le {\left[c_0(r_k+1)\right]}^{\frac{1}{2(r_k+1)}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n\right|}^{2(r_k+1)}\text{d}x}\right)}^{\frac{1}{2(r_k+1)}}\\ \le {\displaystyle \prod _{i=0}^k[}{c}_0(r_i+1){]}^{\frac{1}{2(r_i+1)}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n\right|}^{2(r_0+1)}\text{d}x}\right)}^{\frac{1}{2(r_0+1)}}\\ ={\displaystyle \prod _{i=0}^k[}{c}_0(r_i+1){]}^{\frac{1}{2(r_i+1)}}{\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\left|{\chi }_n\right|}^2\text{d}x}\right)}^{\frac{1}{2}}\end{array}$$

Note that ${\displaystyle {\prod }_{i=0}^k[}{c}_0(r_i+1){]}^{\frac{1}{2(r_i+1)}}=\exp \left\{{\displaystyle {\sum }_{i=0}^k\frac{\ln c_0+i\ln \left(\frac{2^{*}}{2}\right)}{2{\left(\frac{2^{*}}{2}\right)}^i}}\right\}$ is convergent as $k\to \infty$ . We obtain ${\Vert {\chi }_n\Vert }_{L^{\infty }}\le c{\Vert {\chi }_n\Vert }_{L^2}\le c{\Vert u_n\Vert }_{L^2}$ for some constant $c>0$ . Consequently, ${\Vert {\chi }_n\Vert }_{L^{\infty }}\le \frac{l}{2\sqrt{2}}$ for large $n$ because of $(u_n,v_n)\to (0,0)$ in $E$ . Hence

$${\Vert u_n\Vert }_{L^{\infty }}\le {\Vert w_n\Vert }_{L^{\infty }}+{\Vert {\chi }_n\Vert }_{L^{\infty }}\le \frac{l}{\sqrt{2}}$$

for large $n$ . Similarly, ${\Vert v_n\Vert }_{L^{\infty }}\le \frac{l}{\sqrt{2}}$ for large $n$ , which concludes

$${\Vert (u_n,v_n)\Vert }_{L^{\infty }}\le {\left({\Vert u_n\Vert }_{L^{\infty }}^2+{\Vert u_n\Vert }_{L^{\infty }}^2\right)}^{\frac{1}{2}}\le l$$

for large $n$ . The proof is completed.

Proof of Theorem 1. Obviously, the functional $I_h\in C^1\left(E,\mathbb{R}\right)$ , $I_h$ is even under the assumption $\left(f_3\right)$ . By $\left(f_1\right)$ , we can choose $l$ small enough such that

$$\left|F_h(x,u,v)\right|\le {\displaystyle {\int }_0^1\left(\left|u{F}_u(x,tu,tv)\right|+\left|v{F}_v\left(x,tu,tv\right)\right|\right)}\text{d}t\le \frac{1}{p}\beta (x)|(u,v){|}^p+\frac{a_1}{4}|(u,v){|}^2$$

(4)

for all $(x,u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^2$ , which shows

$$I_h(u,v)\ge \frac{1}{4}\min \{a,c,1\}{\Vert (u,v)\Vert }^2-\frac{1}{p}{\int }_{{\mathrm{ℝ}}^N}\beta \left(x\right)\left|\right(u,v){|}^p\text{d}x$$

$$\ge \frac{1}{4}\min \{a,c,1\}{\Vert (u,v)\Vert }^2-\frac{1}{p}{\Vert \beta \Vert }_{\frac{2}{2-p}}{\Vert (u,v)\Vert }_2^p$$

$$\ge \frac{1}{4}\min \{a,c,1\}{\Vert (u,v)\Vert }^2-C{\Vert (u,v)\Vert }^p$$

where $C>0$ is a constant. It can be deduced from the above inequality that $I_h$ is bounded from below and any PS sequence of $I_h$ is bounded in $E$ . Furthermore, $I_h$ satisfies the PS condition.

Now, we verify $\left(A_2\right)$ in Theorem 2. Indeed, for each $k\in \mathbb{N}$ , let $E_k$ be a $k$ dimensional subspace of $E$ . Then there exist $k$ dimensional subspaces $X_k$ of $X$ such that $E_k\subset X_k\times X_k$ . Since all norms are equivalent in $X_k$ there exists a constant ${\tau }_k>0$ such that

$$a_1^{\frac{1}{2}}{\Vert u\Vert }_2\le {\Vert u\Vert }_X\le {\tau }_k{\Vert u\Vert }_2$$

for all $u\in X_k$ .

Let ${\Sigma }_{n_k}=\left\{x\in {\mathbb{R}}^N:\zeta (x)h(|(u,v)|)|(u,v){|}^{\sigma }\ge n_k^{\sigma }{\Vert (u,v)\Vert }_2^2\right\}$ and $\kappa =\frac{1}{2}\max \{a,b,c,d,1\}$ . By Lemma 4, there exists a positive integer $n_k>{\left(2\kappa \sigma {\tau }_k^2\right)}^{\frac{1}{\sigma -1}}$ such that $\text{meas}\left({\Sigma }_{n_k}\right)\ge \frac{1}{n_k}$ for any $(u,v)\in E_k\backslash \{(0,0)\}$ with ${\Vert (u,v)\Vert }_2<{\left(\frac{1}{n_k}\right)}^{\frac{\sigma +1}{2-\sigma }}$ . Consequently

$$\begin{array}{l}{I}_h(u,v)\le \frac{1}{2}\max \{a,b,c,d,1\}{\Vert (u,v)\Vert }^2-{\displaystyle {\int }_{{\mathbb{R}}^N}{F}_h}(x,u,v)\text{d}x\\ \le \kappa {\Vert (u,v)\Vert }^2-\frac{1}{\sigma }{\displaystyle {\int }_{{\mathbb{R}}^N}\zeta }(x)h\left(\left|(u,v)\right|\right)|(u,v){|}^{\sigma }\text{d}x\\ \le \kappa {\Vert (u,v)\Vert }^2-\frac{1}{\sigma }{n}_k^{\sigma -1}{\Vert (u,v)\Vert }_2^2\\ \le \left(\kappa -\frac{1}{\sigma }{n}_k^{\sigma -1}{\tau }_k^{-2}\right){\Vert (u,v)\Vert }^2\le -\kappa {\Vert (u,v)\Vert }^2\end{array}$$

for $(u,v)\in E_k$ with $\Vert (u,v)\Vert \le 1$ and ${\Vert (u,v)\Vert }_2<{\left(\frac{1}{n_k}\right)}^{\frac{\sigma +1}{2-\sigma }}$ . Thus, choosing $0<l_k<\min \left\{1,a_1^{\frac{1}{2}}{\left(\frac{1}{n_k}\right)}^{\frac{\sigma +1}{2-\sigma }}\right\}$ and $A_k=\{(u,v)\in E_k:\Vert (u,v)\Vert =l_k\}$ , we have $\gamma (A_k)\ge k$ and $\underset{u\in A_k}{\sup }{I}_h(u,v)\le -\kappa l_k^2<0$ .

In view of Theorem 2, $I_h$ has a critical point sequence $\left\{\left(u_k,v_k\right)\right\}$ such that $(u_k,v_k)\to (0,0)$ in $E$ and $I_h(u_k,v_k)\le 0$ . By Lemma 5, up to a subsequence, $I_h\left(u_k,v_k\right)=I\left(u_k,v_k\right)$ for every $k$ and $\left\{\left(u_k,v_k\right)\right\}$ is a critical point sequence of $I$ . The proof is completed.

Conclusions

Low energy oscillators arise in many practical applications, which have been ignored by many researchers. Our paper shows that the low energy can induce various solutions, which might be dangerous to a structure or a machine, or a system. This paper sheds a new light on how to deal with small solutions of such systems.