本文将研究如下$p$-Kirchhoff型方程组

$\left\{ \begin{array}{ll} -M_{1}\bigg(\int_{{\Bbb R}^{N}}|\nabla u|^{p}{\rm d}x\bigg)\Delta_{p}u+V(x)|u|^{p-2}u=F_{u}(x, u, v), &x\in{\Bbb R}^{N}, \\ -M_{2}\bigg(\int_{{\Bbb R}^{N}}|\nabla v|^{p}{\rm d}x\bigg)\Delta_{p}v+V(x)|v|^{p-2}v=F_{v}(x, u, v), &x\in{\Bbb R}^{N}, \\ u(x)\rightarrow 0, \;\;\ v(x)\rightarrow 0, \;\;\;\;\ |x|\rightarrow\infty, \end{array}\right.$

(1.1)

其中$M_{1}(s)=a+bs$, $M_{2}(s)=c+ds$ ($a$, $c>0$, $b$, $d\geq 0$), $2\leq p＜N$, $V\in C({\Bbb R}^{N}, {\Bbb R}), $ $F\in C^{1}({\Bbb R}^{N}\times {\Bbb R}^{2}, {\Bbb R})$且$F_{u}=\frac{\partial F}{\partial u}$, $F_{v}=\frac{\partial F}{\partial v}.$

近年来, Kirchhoff型问题引起了人们广泛的关注.这一问题与方程

$\rho \frac{\partial ^{2}u}{\partial t^{2}}-\bigg(\frac{\rho_{0}}{h}+\frac{E}{2L}\int_{0}^{L}\bigg|\frac{\partial u}{\partial x}\bigg|^{2}{\rm d}x\bigg)\frac{\partial ^{2}u}{\partial x^{2}}=0 $

(1.2)

的稳态形式密切相关, 其中$\rho$, $\rho_{0}$, $h$, $E$和$L$为常数.问题(1.2)作为由弹性弦的横向自由振动所产生的古典D'Alembert波方程的一种扩展形式, 由Kirchhoff在文献[1]中最早提出.由于此类问题包含$[0, L]$上的积分, 不再点点恒等, 故称为非局部问题.

随后, 出现了很多Kirchhoff方程模型, 参见文献[2, 7-9, 12-16]等.特别地, Zhou, Wu和Wu在文献[3]中研究了问题(1.1), 其中$p=2$, 且函数$V$和$F$满足如下条件:

$(v_{0})$ $V\in C({\Bbb R}^{N}, {\Bbb R})$满足$\inf\limits_{x\in {\Bbb R}^{N}} V(x)\geq a_{1}>0$, 且对任意$M>0$, 有$\mbox{meas}\{x\in {\Bbb R}^{N}: V(x)\leq M\}＜+\infty$, 其中$a_{1}$是一常数, $\mbox{meas}$表示${\Bbb R}^{N}$中的Lebesgue测度;

$(f_{1})$ $F\in C^{1}({\Bbb R}^{N}\times{\Bbb R}^{2}, {\Bbb R})$, 且存在$2＜p$, $q＜2^{*}$, 有$|F_{u}(x, u, v)|\leq c(1+|(u, v)|^{p-1})$, $|F_{v}(x, u, v)|\leq c(1+|(u, v)|^{q-1})$, 其中$c$为一正常数, $|(u, v)|=(u^{2}+v^{2})^{\frac{1}{2}}$;

$(f_{2})$当$|(u, v)|\rightarrow 0$时, $F_{u}(x, u, v)=o(|(u, v)|)$且$F_{v}(x, u, v)=o(|(u, v)|)$对任意$x\in {\Bbb R}^{N}$一致成立;

$(f_{3})$当$|(u, v)|\rightarrow +\infty$时, $\frac{F(x, u, v)}{|(u, v)|^{4}}\rightarrow \infty$对任意$x\in {\Bbb R}^{N}$一致成立;

$(f_{4})$存在常数$r>0$使得对任意$x\in {\Bbb R}^{N}$且$|(u, v)|\geq r$, 有

$uF_{u}(x, u, v)+vF_{v}(x, u, v)\geq 4F(x, u, v);$

$(f_{5})$对任意$(x, u, v)\in {\Bbb R}^{N}\times {\Bbb R}^{2}$, 有$F(x, -u, -v)=F(x, u, v)$.

于是, 文献[3]得到如下结果.

定理A  假设条件$(v_{0})$, $(f_{1})-(f_{5})$成立.则问题(1.1) ($p=2$)存在无穷多个高能量解, 即解序列$\{(u_{n}, v_{n})\}$满足$\lim\limits_{n\rightarrow\infty}I(u_{n}, v_{n})=+\infty$.

定理B  若将条件$(f_{1})$和$(f_{2})$用如下条件替换, 则定理A的结论依然成立.

$(f_{6})$ $F\in C^{1}({\Bbb R}^{N}\times {\Bbb R}^{2}, {\Bbb R})$, 且存在$2＜p$, $q＜2^{*}$使得$|F_{u}(x, u, v)|\leq C(|(u, v)|+|(u, v)|^{p-1})$, $|F_{v}(x, u, v)|\leq C(|(u, v)|+|(u, v)|^{q-1})$, 其中$C$为一正常数.

受以上结果的启发, 本文将进一步研究问题(1.1), 其中$p\geq2$.通过减弱文献[3]的条件, 我们推广并完善了其中的结果, 且得到与定理A和B类似的结论.

具体来说, 假设函数$V$和$F$满足如下新的条件

$(V_{0})$ $V\in C({\Bbb R}^{N}, {\Bbb R})$满足$\inf\limits_{x\in {\Bbb R}^{N}} V(x)>-\infty$, 且对任意$M>0$, 存在常数$\varrho >0$使得

$\lim\limits_{|y|\rightarrow\infty}\mbox{meas}(\{x\in {\Bbb R}^{N}: |x-y|\leq \varrho , V(x)\leq M\})=0; $

$(F_{1})$ $F\in C^{1}({\Bbb R}^{N}\times {\Bbb R}^{2}, {\Bbb R})$, 且存在常数$c_{1}$, $c_{2}>0$和$\theta_{1}, $ $\theta_{2}\in (p.p^{*})$使得

$|F_{u}(x, u, v)|\leq c_{1}(|(u, v)|^{p-1}+|(u, v)|^{\theta_{1}-1}), \ \mbox{ 且} \ |F_{v}(x, u, v)|\leq c_{2}(|(u, v)|^{p-1}+|(u, v)|^{\theta_{2}-1}), $

其中$|(u, v)|=(u^{2}+v^{2})^{\frac{1}{2}}$;

($F_{2}$)当$|(u, v)|\rightarrow+\infty$时, $\frac{F(x, u, v)}{|(u, v)|^{2p}}\rightarrow \infty$对任意$x\in {\Bbb R}^{N}$一致成立, 且存在$r>0$使得对任意$(x, u, v)\in {\Bbb R}^{N}\times {\Bbb R}^{2}$且$|(u, v)|\geq r$, 有$F(x, u, v)\geq0$;

($F_{3}$)存在$\mu>0$使得对任意$(x, u, v)\in {\Bbb R}^{N}\times {\Bbb R}^{2}$, 都有

$F(x, u, v)\leq \frac{1}{2p}[uF_{u}(x, u, v)+vF_{v}(x, u, v)]+\mu|(u, v)|^{p}.$

本文主要结论如下.

定理1.1  假设条件$(V_{0})$, $(F_{1})-(F_{3})$和($f_{5}$)成立.则问题(1.1)存在无穷多个非平凡解$\{(u_{n}, v_{n})\}$且当$n\rightarrow\infty$时, $\{(u_{n}, v_{n})\}$满足$I(u_{n}, v_{n})\rightarrow+\infty, $其中$I(u, v)$表示问题(1.1)相对应的能量泛函.

注1.1   显然, 条件($V_{0}$)比($v_{0}$)弱.并且, 若在条件$(F_{1})-(F_{3})$中选取$p=2$, 则易由条件$(f_{6})$和$(f_{4})$推出$(F_{3})$.又由于条件$(f_{6})$比$(f_{1})$和$(f_{2})$弱, 则由条件$(f_{1})$, $(f_{2})$和$(f_{4})$也可推出条件$(F_{3})$.

在建立问题(1.1)的变分环境之前, 我们注意到由条件$(V_{0})$知存在常数$V_{1}>0$使得对任意$x\in {\Bbb R}^{N}$, 都有${\tilde{V}}(x)=V(x)+V_{1}>0$成立.令${\tilde{F}}_{u}(x, u, v)=F_{u}(x, u, v)+V_{1}|u|^{p-2}u$, ${\tilde{F}}_{v}(x, u, v)=F_{v}(x, u, v)+V_{1}|v|^{p-2}v$, 且考虑如下新的方程组

$\left\{ \begin{array}{ll} -M_{1}\bigg(\int_{{\Bbb R}^{N}}|\nabla u|^{p}{\rm d}x\bigg)\Delta_{p}u+{\tilde{V}}(x)|u|^{p-2}u={\tilde{F}}_{u}(x, u, v), &x\in{\Bbb R}^{N}, \\[3mm] -M_{2}\bigg(\int_{{\Bbb R}^{N}}|\nabla v|^{p}{\rm d}x\bigg)\Delta_{p}v+{\tilde{V}}(x)|v|^{p-2}v={\tilde{F}}_{v}(x, u, v), &x\in{\Bbb R}^{N}, \\[2mm] u(x)\rightarrow 0, \;\;\;\ v(x)\rightarrow 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\ |x|\rightarrow\infty. \end{array}\right.$

(2.1)

则易得问题(1.1)和(2.1)等价.事实上, 若条件$(V_{0})$, $(F_{1})-(F_{3})$和$(f_{5})$对函数$V$, $F_{u}$和$F_{v}$成立, 则对函数${\tilde{V}}$, ${\tilde{F}}_{u}$和${\tilde{F}}_{v}$也成立.

下面, 我们只需研究等价问题(2.1).于是, 可作如下假设

$({\tilde{V}}_{0})$ $V\in C({\Bbb R}^{N}, {\Bbb R})$满足$\inf\limits_{x\in {\Bbb R}^{N}} V(x)>0$, 且对任意$M>0$, 存在常数$\varrho >0$使得

$\lim\limits_{|y|\rightarrow\infty}\mbox{meas}(\{x\in {\Bbb R}^{N}: |x-y|\leq \varrho , V(x)\leq M\})=0.$

现在, 引入空间

$E:=\bigg\{u\in W^{1, p}({\Bbb R}^{N}): \int_{{\Bbb R}^{N}}V(x)|u|^{p}{\rm d}x＜+\infty\bigg\}, $

且赋有范数

$\|u\|=\bigg(\int_{{\Bbb R}^{N}}( |\nabla u|^{p}+V(x)|u|^{p}){\rm d}x\bigg)^{\frac{1}{p}}.$

显然, $X=E\times E$是一自反的Banach空间, 且$X$中的范数定义为

$\|(u, v)\|=(\|u\|^{p}+\|v\|^{p})^{\frac{1}{p}}.$

显然, $X$可连续嵌入到$W^{1, p}({\Bbb R}^{N})\times W^{1, p}({\Bbb R}^{N})$, 且对任意$p\leq \sigma＜p^{*}$, $X$可连续嵌入到$L^{\sigma}({\Bbb R}^{N})\times L^{\sigma}({\Bbb R}^{N})$, 即存在$\gamma_{\sigma}>0$使得对任意$(u, v)\in X$, 有

$\|(u, v)\|_{\sigma}\leq \gamma_{\sigma}\|(u, v)\|, $

(2.2)

其中$\|(u, v)\|_{\sigma}=(\int_{{\Bbb R}^{N}}(|u|^{\sigma}+|v|^{\sigma}){\rm d}x)^{\frac{1}{\sigma}}$.

由文献[11]中的引理3.1, 我们可进一步得到如下结果.

引理2.1  假设条件$({\tilde{V}}_{0})$成立.则对任意$p\leq \sigma＜p^{*}$, 嵌入$X\hookrightarrow L^{\sigma}({\Bbb R}^{N})\times L^{\sigma}({\Bbb R}^{N})$是紧的.

对任意$(u, v)\in X$, 考虑能量泛函

$I(u, v)=\frac{a}{p}\int_{{\Bbb R}^{N}}|\nabla u|^{p}{\rm d}x+\frac{b}{2p}\bigg(\int_{{\Bbb R}^{N}}|\nabla u|^{p}{\rm d}x\bigg)^{2}+\frac{c}{p}\int_{{\Bbb R}^{N}}|\nabla v|^{p}{\rm d}x+ \frac{d}{2p}\bigg(\int_{{\Bbb R}^{N}}|\nabla v|^{p}{\rm d}x\bigg)^{2}\\ \quad \quad \quad +\frac{1}{p}\int_{{\Bbb R}^{N}}V(x)|u|^{p}{\rm d}x+\frac{1}{p}\int_{{\Bbb R}^{N}}V(x)|v|^{p}{\rm d}x-\int_{{\Bbb R}^{N}}F(x, u, v){\rm d}x.$

由条件$(F_{1})$知, 对任意$(x, u, v)\in {\Bbb R}^{N}\times {\Bbb R}^{2}$, 有

$|F(x, u, v)|\leq C(|(u, v)|^{p}+|(u, v)|^{\theta_{1}}+|(u, v)|^{\theta_{2}}). $

(2.3)

因此, 由假设$({\tilde{V}}_{0})$和$(F_{1})$可知泛函$I\in C^{1}(X, {\Bbb R})$且对任意$\forall (\varphi , \psi)\in X$, 有

$\langle I'(u, v), (\varphi , \psi)\rangle\\ =a\int_{{\Bbb R}^{N}}|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi {\rm d}x+b\int_{{\Bbb R}^{N}}|\nabla u|^{p}{\rm d}x\int_{{\Bbb R}^{N}}|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi {\rm d}x\\+c\int_{{\Bbb R}^{N}}|\nabla v|^{p-2}\nabla v\cdot \nabla \psi {\rm d}x+ d\int_{{\Bbb R}^{N}}|\nabla v|^{p}{\rm d}x\int_{{\Bbb R}^{N}}|\nabla v|^{p-2}\nabla v\cdot \nabla \psi {\rm d}x\\ +\int_{{\Bbb R}^{N}}V(x)(|u|^{p-2}u\varphi +|v|^{p-2}v\psi) {\rm d}x -\int_{{\Bbb R}^{N}}\big{(}\varphi F_{u}(x, u, v)+\psi F_{v}(x, u, v)\big{)}{\rm d}x.$

并且, 泛函$I$在$X$中的临界点即为问题(2.1)的弱解.问题(2.1)的弱解是指存在$(u, v)\in X$满足对任意$(\varphi , \psi)\in X$, 有$\langle I'(u, v), (\varphi , \psi)\rangle=0$.

下面, 给出文献[10]中的对称山路引理的具体内容, 这是本章定理证明的主要依据.

命题2.1 ^{[10, 定理9.12]}  令$E$是一无限维的实Banach空间, 且$E=Y\oplus Z$, 其中$Y$有限维.若对任意$c>0$, $I\in C^{1}(E, {\Bbb R})$满足$(PS)_{c}$条件, 且有

$(I_{1})$ $I(0)=0$, 且对任意$u\in E$, 有$I(-u)=I(u)$;

$(I_{2})$存在常数$\rho$, $\alpha >0$使得$I|_{\partial B_{\rho}\cap Z}\geq \alpha $;

$(I_{3})$对任意有限维子空间${\tilde{E}}\subset E$, 存在$R=R({\tilde{E}})>0$使得对任意$u\in{\tilde{E}}\setminus B_{R}$, 有$I(u)\leq 0$.

则泛函$I$存在一个无界的临界值序列.

引理2.2   假设条件$({\tilde{V}}_{0})$, $(F_{1})-(F_{3})$成立.则对任意$c>0$, 泛函$I$满足$(PS)_{c}$条件.

证   令$\{(u_{n}, v_{n})\}\subset X$为泛函$I$的任一$(PS)_{c}$序列.则有

$I(u_{n}, v_{n})=c+o(1), \;\ I'(u_{n}, v_{n})=o(1).$

(2.4)

首先, 证明$\{(u_{n}, v_{n})\}$在$X$中有界.反证.假设当$n\rightarrow\infty$时, $\|(u_{n}, v_{n})\|\rightarrow\infty$, 令$(w_{n}, z_{n})=(\frac{u_{n}}{\|(u_{n}, v_{n})\|}, \frac{v_{n}}{\|(u_{n}, v_{n})\|})$.显然, $\|(w_{n}, z_{n})\|=1$在$X$中有界.由引理2.1知存在子序列(仍记为$\{(w_{n}, z_{n})\}$)和$(w, z)\in X$使得

$(w_{n}, z_{n})\rightharpoonup (w, z)\;\ \mbox{于}\;\ X, $

$(w_{n}, z_{n})\rightarrow (w, z)\;\ \mbox{于}\;\ L^{\sigma}({\Bbb R}^{N})\times L^{\sigma}({\Bbb R}^{N}), \;\ p\leq \sigma＜p^{*}, $

$(w_{n}, z_{n})\rightarrow (w, z)\;\ \mbox{a.e.}\;\ \mbox{于}\ {\Bbb R}^{N}.$

若$(w, z)=(0, 0)$, 由(2.4)式, 条件$(F_{3})$和Minkowski不等式, 易得

$\frac{c+1+o(1)}{\|(u_{n}, v_{n})\|^{p}} \ge \frac{1}{\|(u_{n}, v_{n})\|^{p}}\bigg{[}I(u_{n}, v_{n})-\frac{1}{2p}\langle I'(u_{n}, v_{n}), (u_{n}, v_{n})\rangle\bigg{]}\\ \quad \quad \quad =\frac{1}{\|(u_{n}, v_{n})\|^{p}}\bigg{\{}\frac{a}{2p}\int_{{\Bbb R}^{N}}|\nabla u_{n}|^{p}{\rm d}x+\frac{c}{2p}\int_{{\Bbb R}^{N}}|\nabla v_{n}|^{p}{\rm d}x\\ \quad \quad \quad +\frac{1}{2p}\int_{{\Bbb R}^{N}}V(x)(|u_{n}|^{p} +|v_{n}|^{p}){\rm d}x\\ \quad \quad \quad +\int_{{\Bbb R}^{N}}\bigg{[}\frac{1}{2p}(u_{n}F_{u}(x, u_{n}, v_{n}) +v_{n}F_{v}(x, u_{n}, v_{n}))-F(x, u_{n}, v_{n})\bigg{]}{\rm d}x\bigg{\}}\\ \quad \quad \quad \ge \frac{1}{2p}\min\{a, c, 1\}-\mu\int_{{\Bbb R}^{N}}|(w_{n}, z_{n})|^{p}{\rm d}x\\ \quad \quad \quad \ge \frac{1}{2p}\min\{a, c, 1\}-2^{\frac{p}{2}-1}\mu\|(w_{n}, z_{n})\|_{p}^{p},$

即得到$0\geq \frac{1}{2p}\min\{a, c, 1\}$, 矛盾.

若$(w, z)\neq (0, 0)$, 对任意$0\leq\alpha _{1}＜\alpha _{2}$, 我们考虑

$\Lambda_{n}(\alpha _{1}, \alpha _{2}):=\{x\in {\Bbb R}^{N}:\alpha _{1}\leq |(u_{n}(x), v_{n}(x))|＜\alpha _{2}\}$

和

$A:=\{x\in{\Bbb R}^{N}: (w(x), z(x))\neq (0, 0)\}.$

则有$\mbox{meas}(A)>0$且对$x\in A$, 有$\lim\limits_{n\rightarrow\infty}|(u_{n}, v_{n})|=+\infty$几乎处处成立.于是, 当$n$充分大时, $A\subset \Lambda_{n}(r, +\infty)$, 其中$r$由条件$(F_{2})$给出.因此, 利用(2.3)和(2.4)式, 条件$(F_{2})$和Fatou引理, 可得

$\begin{eqnarray*} 0&=&\lim\limits_{n\rightarrow\infty}\frac{c+o(1)}{\|(u_{n},v_{n})\|^{2p}}=\lim\limits_{n\rightarrow\infty}\frac{I(u_{n},v_{n})}{\|(u_{n},v_{n})\|^{2p}}\\ &\leq& \frac{1}{p}\max\{a,c,1\}\lim\limits_{n\rightarrow\infty}\frac{1}{\|(u_{n},v_{n})\|^{p}}+\frac{1}{2p}\max\{b,d\}\\&&-\lim\limits_{n\rightarrow\infty} \int_{\Lambda_{n}(0,r)}\frac{F(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|^{2p}}{\rm d}x -\lim\limits_{n\rightarrow\infty}\int_{\Lambda_{n}(r,+\infty)}\frac{F(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|^{2p}}{\rm d}x\\ &\leq& \frac{1}{p}\max\{a,c,1\}\lim\limits_{n\rightarrow\infty}\frac{1}{\|(u_{n},v_{n})\|^{p}}+\frac{1}{2p}\max\{b,d\} -\lim\limits_{n\rightarrow\infty}\int_{A}\frac{F(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|^{2p}}{\rm d}x\\&&+C\lim\limits_{n\rightarrow\infty}\int_{\Lambda_{n}(0,r)}\frac{1}{\|(u_{n},v_{n})\|^{2p}} \mbox{[}|(u_{n},v_{n})|^{p}+|(u_{n},v_{n})|^{\theta_{1}}+|(u_{n},v_{n})|^{\theta_{2}}\mbox{]}{\rm d}x\\ &\leq&\frac{1}{p}\max\{a,c,1\}\lim\limits_{n\rightarrow\infty}\frac{1}{\|(u_{n},v_{n})\|^{p}}+\frac{1}{2p}\max\{b,d\} \\&&-\lim\limits_{n\rightarrow\infty}\int_{A}\frac{F(x,u_{n},v_{n})}{|(u_{n},v_{n})|^{2p}}|(w_{n},z_{n})|^{2p}{\rm d}x\\ &&+C(1+r^{\theta_{1}-p}+r^{\theta_{2}-p})\lim\limits_{n\rightarrow\infty}\int_{\Lambda_{n}(0,r)} \frac{|(w_{n},z_{n})|^{p}}{\|(u_{n},v_{n})\|^{p}}{\rm d}x\rightarrow-\infty, \end{eqnarray*}$

矛盾.故得$\{(u_{n}, v_{n})\}$在$X$中有界.于是, 存在子序列(仍记为$\{(u_{n}, v_{n})\}$)和$(u, v)\in X$使得

$(u_{n}, v_{n})\rightharpoonup (u, v)\;\ \mbox{于}\;\ X, $

$(u_{n}, v_{n})\rightarrow (u, v)\;\ \mbox{于}\;\ L^{\sigma}({\Bbb R}^{N})\times L^{\sigma}({\Bbb R}^{N}), \;\ p\leq \sigma＜p^{*}.$

利用基本不等式

$\langle|x|^{p-2}x-|y|^{p-2}y, x-y\rangle\geq C_{p}|x-y|^{p}, $

有

$\begin{eqnarray*} \min\{a,1\}\|u_{n}-u\|^{p}&\leq& \langle I'(u_{n},v_{n})-I'(u,v),(u_{n}-u,0)\rangle\\ &&-b\bigg(\int_{{\Bbb R}^{N}}(|\nabla u_{n}|^{p}-|\nabla u|^{p}){\rm d}x\bigg)\int_{{\Bbb R}^{N}}|\nabla u|^{p-2}\nabla u\cdot \nabla (u_{n}-u){\rm d}x\\&&+\int_{{\Bbb R}^{N}}(F_{u}(x,u_{n},v_{n})-F_{u}(x,u,v))(u_{n}-u){\rm d}x. \end{eqnarray*}$

由于$\{u_{n}\}$在$E$中有界且根据弱收敛的定义, 当$n\rightarrow\infty$时, 易得

$b\bigg(\int_{{\Bbb R}^{N}}(|\nabla u_{n}|^{p}-|\nabla u|^{p}){\rm d}x\bigg)\int_{{\Bbb R}^{N}}|\nabla u|^{p-2}\nabla u\cdot \nabla (u_{n}-u){\rm d}x\rightarrow 0.$

由条件$(F_{1})$和Hölder不等式知当$n\rightarrow\infty$时, 有

$\begin{eqnarray*} &&\int_{{\Bbb R}^{N}}|F_{u}(x,u_{n},v_{n})-F_{u}(x,u,v)||u_{n}-u|{\rm d}x\\&\leq&C\int_{{\Bbb R}^{N}}\big{[}|(u_{n},v_{n})|^{p-1}+|(u,v)|^{p-1}+|(u_{n},v_{n})|^{\theta_{1}-1}+|(u,v)|^{\theta_{1}-1}\\&&+|(u_{n},v_{n})|^{\theta_{2}-1}+ |(u,v)|^{\theta_{2}-1}\big{]}|u_{n}-u|{\rm d}x\\ &\leq&C\big{[}(|(u_{n},v_{n})|_{p}^{p-1}+|(u,v)|_{p}^{p-1})\|u_{n}-u\|_{p}\\&&+(|(u_{n},v_{n})|_{\theta_{1}}^{\theta_{1}-1}+|(u,v)|_{\theta_{1}}^{\theta_{1}-1})\|u_{n}-u\|_{\theta_{1}}\\ &&+(|(u_{n},v_{n})|_{\theta_{2}}^{\theta_{2}-1}+|(u,v)|_{\theta_{2}}^{\theta_{2}-1})\|u_{n}-u\|_{\theta_{2}}\big{]}\\ &\leq& C(\|u_{n}-u\|_{p}+\|u_{n}-u\|_{\theta_{1}}+\|u_{n}-u\|_{\theta_{2}})\rightarrow0, \end{eqnarray*}$

其中$|(u, v)|_{\sigma}=(\int_{{\Bbb R}^{N}}|(u, v)|^{\sigma}{\rm d}x)^{\frac{1}{\sigma}}$.故由$I'(u_{n}, v_{n})\rightarrow 0$且$u_{n}\rightharpoonup u$于$E$, 可得$u_{n}\rightarrow u$于$E$.类似地, 可证$v_{n}\rightarrow v$于$E$.证毕.

令$E^{*}$表示$E$的对偶子空间且$\langle\cdot\rangle$表示$E^{*}$与$E$之间的对偶积.由于$E$是一自反且可分的Banach空间, 故存在$e_{j}\in E$和$e_{j}^{*}\in E^{*} $ $(j=1, 2, \cdots )$满足:

(ⅰ) $\langle e_{i}, e_{j}^{*}\rangle=\delta _{i, j}$, 其中$\delta _{i, j}=1$, $i=j$, $\delta _{i, j}=0$, $i\neq j$;

(ⅱ) $E=\overline{\mbox{span}\{e_{1}, e_{2}, \cdots \}}$, $E^{*}=\overline{\mbox{span}\{e_{1}^{*}, e_{2}^{*}, \cdots \}}.$

令$E_{i}=\mbox{span}\{e_{i}\}$, $Y_{k} = %\underset{i=1}{\overset{k} \bigoplus\limits^k_{i=1} E_{i}$, $Z_{k}= \overline{\bigoplus\limits^\infty_{i=k+1} E_{i}}$.则有$E=Y_{k}\bigoplus Z_{k}$且$X=(Y_{k}\times Y_{k})\bigoplus (Z_{k}\times Z_{k}).$定义

$\beta_{k}(\sigma):= \sup\limits_{(u, v)\in Z_{k}\times Z_{k}, \atop \|(u, v)\|=1} |(u, v)|_{\sigma}, \;\;\ \sigma\in[p, p^{*}).$

则有如下引理.

引理3.1  假设条件$({\tilde{V}}_{0})$成立.则对任意$p\leq\sigma＜p^{*}$, 当$k\rightarrow\infty$时, 有$\beta_{k}(\sigma)\rightarrow0$.

证  由于$0\leq \beta_{k+1}\leq \beta_{k}$, 当$k\rightarrow\infty$时, 有$\beta_{k}\rightarrow\beta_{0}\geq 0$.若$\beta_{0}>0$, 由$\beta_{k}$的定义, 存在$(u_{k}, v_{k})\in Z_{k}\times Z_{k}$且$\|(u_{k}, v_{k})\|=1$满足对所有的$k\geq1$, 都有$-\frac{1}{k}\leq \beta_{0}-|(u_{k}, v_{k})|_{\sigma}\leq\frac{1}{k}$.于是, 存在子序列(仍记为$\{(u_{k}, v_{k})\}$)使得$u_{k}\rightharpoonup u$, $v_{k}\rightharpoonup v$于$E$, 且对所有的$j\geq1$, 有$\langle u, e_{j}^{*}\rangle=\lim\limits_{k\rightarrow\infty}\langle u_{k}, e_{j}^{*}\rangle=0$.故可得$u=0$, 即有$u_{k}\rightharpoonup 0$于$E$.从而, 对任意$\sigma\in [p, p^{*})$, 有$u_{k}\rightarrow 0$于$L^{\sigma}({\Bbb R}^{N})$.

同理可证对任意$\sigma\in [p, p^{*})$, $v_{k}\rightarrow 0$于$L^{\sigma}({\Bbb R}^{N})$.矛盾.因此, $\beta_{0}=0.$证毕.

利用引理3.1, 选取正整数$m\geq1$满足对任意$(u, v)\in Z_{m}\times Z_{m}$, 有

$|(u, v)|_{p}^{p}\leq \beta_{k}^{p}(p)\|(u, v)\|^{p}, \;\ |(u, v)|_{\theta_{j}}^{\theta_{j}}\leq \beta_{k}^{\theta_{j}}(\theta_{j})\|(u, v)\|^{\theta_{j}}, \;\ j=1, 2 . $

(3.1)

引理3.2  假设条件$({\tilde{V}}_{0})$, $(F_{1})$成立.则存在常数$\rho, $ $\alpha >0$使得$I|_{\partial B_{\rho}\cap (Z_{m}\times Z_{m})}\geq\alpha .$

证  对任意$(u, v)\in Z_{m}\times Z_{m}$, 利用(2.3)和(3.1)式, 有

$I(u, v) \ge \frac{1}{p}\min\{a, c, 1\}\|(u, v)\|^{p}-C\big{[}|(u, v)|_{p}^{p}+|(u, v)|_{\theta_{1}}^{\theta_{1}}+|(u, v)|_{\theta_{2}}^{\theta_{2}}\big{]}\\ \quad \quad \quad \ge \big{[}\frac{1}{p}\min\{a, c, 1\}-C\beta_{k}^{p}(p)\big{]}\|(u, v)\|^{p}\\ \quad \quad \quad \quad -C\beta_{k}^{\theta_{1}}(\theta_{1})\|(u, v)\|^{\theta_{1}}-C\beta_{k}^{\theta_{2}}(\theta_{2})\|(u, v)\|^{\theta_{2}}.$

由于$\beta_{k}^{p}(p)\rightarrow 0$, 故可选取常数$k_{0}>0$使得$C\beta_{k_{0}}^{p}(p)＜\frac{1}{2p}\min\{a, c, 1\}.$因此, 若$(u, v)\in Z_{m}\times Z_{m}$且$\|(u, v)\|=\rho$充分小, 可得$I(u, v)\geq\alpha >0.$证毕.

引理3.3  假设条件$({\tilde{V}}_{0})$, $(F_{1})$和$(F_{2})$成立.则对任意有限维子空间${\tilde{X}}\subset X$, 存在$R=R({\tilde{X}})>0$使得对任意$(u, v)\in{\tilde{X}}\backslash B_{R}, $有$I(u, v)\leq0$.

证  对任意有限维子空间${\tilde{X}}\subset X$, 存在一有限维子空间${\tilde{E}}\subset E$使得${\tilde{X}}\subset {\tilde{E}}\times {\tilde{E}}$.并且, 由于${\tilde{E}}$中的范数等价, 存在$\tau>0$满足

$\|u\|_{2p}\geq \tau \|u\|. $

(3.2)

由条件$(F_{1})$和$(F_{2})$知对任意$M>\frac{\max\{b, d\}}{2p\tau^{2p}}$, 存在$C_{M}>0$, 使得对任意$(x, u, v)\in {\Bbb R}^{N}\times {\Bbb R}^{2}$, 有

$F(x, u, v)\geq M|(u, v)|^{2p}-C_{M}|(u, v)|^{p}. $

(3.3)

于是, 利用(2.3), (3.2), (3.3)式和Minkowski不等式, 有

$\begin{eqnarray*} I(u,v)&\leq& \frac{1}{p}\max\{a,c,1\}\|(u,v)\|^{p}+\frac{b}{2p}\|u\|^{2p}+\frac{d}{2p}\|v\|^{2p}\\&&-M(\|u\|_{2p}^{2p}+\|v\|_{2p}^{2p})+2^{\frac{p}{2}-1}C_{M}\|(u,v)\|_{p}^{p}\\ &\leq& \big{[}\frac{1}{p}\max\{a,c,1\}+2^{\frac{p}{2}-1}C_{M}\gamma_{p}^{p}\big{]}\|(u,v)\|^{p}\\&&-(M\tau^{2p}-\frac{b}{2p})\|u\|^{2p}-(M\tau^{2p}-\frac{d}{2p})\|v\|^{2p}\\ &\leq& \big{[}\frac{1}{p}\max\{a,c,1\}+2^{\frac{p}{2}-1}C_{M}\gamma_{p}^{p}\big{]}\|(u,v)\|^{p}\\&&-\frac{1}{2}\min \{(M\tau^{2p}-\frac{b}{2p}),(M\tau^{2p}-\frac{d}{2p})\}\|(u,v)\|^{2p}. \end{eqnarray*}$

故存在充分大的$R=R({\tilde{X}})>0$使得对任意$(u, v)\in{\tilde{X}}\backslash B_{R}$, 有$I(u, v)\leq0$.

定理1.1的证明  显然, $I(0, 0)=0$, 且由条件$(f_{5})$知$I$是偶泛函.结合引理2.2, 3.2和3.3知命题2.1的条件都满足.故问题(2.1)存在无穷多个非平凡的解$\{(u_{n}, v_{n})\}$且当$n\rightarrow\infty$时, $I(u_{n}, v_{n})\rightarrow \infty$, 即问题(1.1)存在无穷多个高能量解.