近些年来, 随着天体物理学、化学和生物科学等诸多学科的发展, 涌现出了大量的二阶非线性椭圆方程. 含有非局部项的椭圆方程作为一类特殊的方程, 受到了数学家的广泛关注. Chen等^[1]运用对称山路引理研究了具有凹凸非线性$ p $-Kirchhoff型椭圆问题

$ \Big (a+\mu\Big(\int_{{{\Bbb R}} ^{N}}(|\nabla u|^{p}+V(x)|u|^{p}){\rm d}x\Big)^\tau\Big)(-\Delta_{p}u+V(x)|u|^{p-2}u)=f(x, u), x\in{{\Bbb R}} ^{N}\nonumber $

无穷多解的存在性. Yan等^[2]运用不动点指标理论得到了一类非局部椭圆边值问题

$ \left\{\begin{array}{ll} { } -a\Big(\int_\Omega |u(x)|^r{\rm d}x\Big) \Delta u=\lambda f(x, u), &x\in \Omega, \nonumber\\ u(x)>0, & x\in \Omega, \nonumber\\ u(x)=0, &x\in \partial \Omega\nonumber \end{array}\right. $

多个正解的存在性. Lu等^[3]运用不动点指标理论研究了一类非局部椭圆边值问题

$ \left\{\begin{array}{ll} L_k u+f(x, u)=0, & x\in \Omega, \nonumber\\ u(x)=0, &x\in \partial \Omega\nonumber \end{array}\right. $

正解的存在性, 其中

$ L_k u(x)=\frac{1}{2}\int_{{{\Bbb R}} ^{N}}(u(x+y)+u(x-y)-2u(x))K(y){\rm d}y, x\in {{\Bbb R}} ^{N}.\nonumber $

受上述文献的启发, 本文研究拟线性椭圆方程组边值问题

(1.1) $ \begin{eqnarray} \left\{\begin{array}{ll} { } M\Big(\int_{{{\Bbb R}} ^{N}}(|\nabla u|^{p}+V(x)|u|^{p}){\rm d}x\Big)(-\Delta_{p}u+V(x)|u|^{p-2}u)=\sigma d^{-1}F_{u}(x, u, v)+\lambda |u|^{q-2}u, \\ [3mm] { } M\Big(\int_{{{\Bbb R}} ^{N}}(|\nabla v|^{p}+V(x)|v|^{p}){\rm d}x\Big)(-\Delta_{p}v+V(x)|v|^{p-2}v)=\sigma d^{-1}F_{v}(x, u, v)+\mu |v|^{q-2}v, \\ u, v\in W^{1, p}({{\Bbb R}} ^{N}), x\in{{\Bbb R}} ^{N} \end{array}\right. \end{eqnarray} $

无穷多解的存在性, 其中$ M(s)=s^{k}, k>0, N\geq3, p>1, p(k+1)<q\leq d<p^{\ast} $ (当$ p<N $时, $ p^{\ast}=\frac{pN}{N-p} $; 当$ p=N $时, $ p^{\ast}=+\infty $), $ \lambda, \mu>0, \sigma\in{{\Bbb R}} ^{N} $.

在本文中, 我们做如下假设:

(H1) $ V(x)\in L^\infty ({{\Bbb R}} ^N) $且存在常数$ V_0, V_1>0 $使得对于任意$ x\in {{\Bbb R}} ^N $有$ V_0\leq V(x)\leq V_1 $;

(H2) $ F(x, u, v)\in C^{1}({{\Bbb R}} ^{N}\times{{\Bbb R}} ^{2}) $是度为$ d $的正齐次函数, 即对任意$ (x, u, v)\in{{\Bbb R}} ^{N}\times {{\Bbb R}} ^{2}, t>0 $有$ F(x, tu, tv)=t^{d}F(x, u, v) $; $ F_{u}(x, u, v) $关于变量$ u $是奇函数, 关于变量$ v $是偶函数; $ F_{v}(x, u, v) $关于变量$ u $是偶函数, 关于变量$ v $是奇函数. 此外, 存在常数$ c_{0}>0 $使得对于任意$ (x, u, v), (x, \xi, \eta)\in{{\Bbb R}} ^{N}\times{{\Bbb R}} ^{2} $有

(1.2) $ \begin{equation} 0\leq F(x, u, v), F_{u}(x, u, v)u, F_{v}(x, u, v)v\leq c_{0}(|u|^{d}+|v|^{d}) \end{equation} $

及

(1.3) $ \begin{equation} |F(x, u, v)-F(x, \xi, \eta)|\leq c_{0}(|u|^{d-1}+|v|^{d-1}+|\xi|^{d-1}+|\eta|^{d-1})(|u-\xi|+|v-\eta|). \end{equation} $

注1.1  若$ F(x, u, v)=|u|^{\alpha}|v|^{\beta}, \alpha, \beta>1, \alpha+\beta=d $则函数$ F(x, u, v) $满足假设条件$ \rm (H2) $.

设$ E\equiv W^{1, p}({{\Bbb R}} ^{N}) $为通常的Sobolev空间, 其上的范数为

$ \|u\|_{E}=\Big(\int_{{{\Bbb R}} ^{N}}(|\nabla u|^p+V(x)|u|^p){\rm d}x\Big)^{1/p}.\nonumber $

由Sobolev嵌入不等式(参见文献[4])知: 当$ 1<p<N, p\leq r\leq p^{*}=\frac{Np}{N-p} $时, 存在常数$ S_{r}>0 $使得

(2.1) $ \begin{equation} \|u\|_{L^{r}({{\Bbb R}} ^{N})}\leq S_{r}\|u\|_{E}, \forall u\in E. \end{equation} $

当$ p=N, m\geq N $时, 存在常数$ S_{m}>0 $使得

(2.2) $ \begin{equation} \|u\|_{L^{m}({{\Bbb R}} ^{N})}\leq S_{m}\|u\|_{E}, \forall u\in E. \end{equation} $

构造积空间$ X=E\times E $, 其上的范数定义为

$ \|(u, v)\|=(\|u\|^{p}_{E}+\|v\|^{p}_{E})^{1/p}, \forall(u, v)\in X, $

则$ X $对此范数构成一自反的Banach空间.

定义2.1  设$ (u, v)\in X $, 如果对于任意的$ (\varphi, \psi)\in X $, 有

$ \begin{eqnarray*} &&\|u\|^{pk}_{E}\int_{{{\Bbb R}} ^{N}}(|\nabla u|^{p-2}\nabla u\nabla \varphi+V(x)|u|^{p-2}u\varphi){\rm d}x\nonumber\\ &&+\|v\|^{pk}_{E}\int_{{{\Bbb R}} ^{N}}(|\nabla v|^{p-2}\nabla v\nabla\psi+V(x)|v|^{p-2}v\psi){\rm d}x\nonumber\\ &=&\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}(F_{u}(x, u, v)\varphi+F_{v}(x, u, v)\psi){\rm d}x +\int_{{{\Bbb R}} ^{N}}(\lambda|u|^{q-2}u\varphi+\mu|v|^{q-2}v\psi){\rm d}x\nonumber \end{eqnarray*} $

成立, 则称函数$ (u, v) $是问题(1.1)的弱解.

设$ J: X\rightarrow {{\Bbb R}} $是问题(1.1)所对应的能量泛函, 其具体定义为

(2.3) $ \begin{equation} J(u, v)=\frac{1}{m}(\|u\|^{m}_{E}+\|v\|^{m}_{E})-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F(x, u, v){\rm d}x -\frac{\lambda}{q}\int_{{{\Bbb R}} ^{N}}|u|^{q}{\rm d}x-\frac{\mu}{q}\int_{{{\Bbb R}} ^{N}}|v|^{q}{\rm d}x, \end{equation} $

其中$ m=p(k+1). $易见, $ J\in C^1(X, {{\Bbb R}} ) $且对于任意的$ (\varphi, \psi)\in X $有

$ \begin{eqnarray*} J'(u, v)(\varphi, \psi)&=&\|u\|^{pk}_{E}\int_{{{\Bbb R}} ^{N}}(|\nabla u|^{p-2}\nabla u\nabla\varphi +V(x)|u|^{p-2}u\varphi){\rm d}x\nonumber\\ &&+\|v\|^{pk}_{E}\int_{{{\Bbb R}} ^{N}}(|\nabla v|^{p-2}\nabla v\nabla\psi+V(x)|v|^{p-2}v\psi){\rm d}x\nonumber\\ &&-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}(F_{u}(x, u, v)\varphi+F_{v}(x, u, v)\psi){\rm d}x -\int_{{{\Bbb R}} ^{N}}(\lambda|u|^{q-2}u\varphi+\mu|v|^{q-2}v\psi){\rm d}x.\nonumber \end{eqnarray*} $

引理2.1  设条件$ \rm (H1)–(H2) $成立. 若$ \{(u_{n}, v_{n})\} $是泛函$ J $在$ X $中的$ (PS)_c $序列, 则$ \{(u_{n}, v_{n})\} $在$ X $中有界.

证  设$ \{(u_{n}, v_{n})\} $是泛函$ J $在$ X $中的$ (PS)_c $序列, 则当$ n\to \infty $时$ J(u_{n}, v_{n})\rightarrow c $, $ J'(u_{n}, v_{n})\rightarrow 0 $. 任意取定$ \theta \in {{\Bbb R}} $, 使得$ p(k+1)<\theta<q\leq d $, 则当$ n $足够大时, 有

(2.4) $ \begin{eqnarray} c+1+\|(u_{n}, v_{n})\|&\geq & J(u_{n}, v_{n})-\frac{1}{\theta}J'(u_{n}, v_{n})(u_{n}, v_{n})\\ &=&(\frac{1}{m}-\frac{1}{\theta})(\|u_{n}\|^{m}_{E}+\|v_{n}\|^{m}_{E}) +\sigma(\frac{1}{\theta}-\frac{1}{d})\int_{{{\Bbb R}} ^{N}}F(x, u_{n}, v_{n}){\rm d}x\\ & & + (\frac{1}{\theta}-\frac{1}{q})\int_{{{\Bbb R}} ^{N}}(\lambda|u_{n}|^{q}+\mu|v_{n}|^{q}){\rm d}x\\ &\geq&(\frac{1}{m}-\frac{1}{\theta})(\|u_{n}\|^{m}_{E}+\|v_{n}\|^{m}_{E})\\ &\geq&(\frac{1}{m}-\frac{1}{\theta})2^{1-\frac{m}{p}}\|(u, v)\|^m, \end{eqnarray} $

其中$ m=p(k+1) $. 由于$ 1<m<\theta $, 从而$ \{(u_{n}, v_{n})\} $在$ X $中有界. 证毕.

设序列$ \{(u_{n}, v_{n})\} $是泛函$ J $在$ X $中的$ (PS)_c $序列, 由引理2.1知序列$ \{(u_{n}, v_{n})\} $在$ X $中有界, 从而存在一正数$ M_0 >0 $使得$ \|(u_n, v_n)\| \leq M_0 . $又由于$ X $是自反的Banach空间, 从而存在$ \{(u_n, v_n)\} $的一个子列(不妨仍记为$ \{(u_n, v_n)\} $), 在$ X $中弱收敛于元$ (u, v)\in X $. 由文献[5]知

(2.5) $ \begin{equation} (u_{n}(x), v_{n}(x))\rightarrow (u(x), v(x)), \ {\rm a.e.}\ x\in{{\Bbb R}} ^{N}. \end{equation} $

类似于文献[6]中引理2.3的证明可得如下引理:

引理2.2  设$ \rm (H1)–(H2) $成立且$ p<q\leq d $. 若$ \{(u_{n}, v_{n})\} $是泛函$ J $在$ X $中的$ (PS) $序列且满足(2.5)式, 对于足够小的$ \epsilon>0, $存在$ r_{0}>1 $使得当$ r>r_{0} $时, 有

(2.6) $ \begin{equation} \limsup\limits_{n\rightarrow \infty}\int_{B^{c}_{r}}|u_{n}|^{q}{\rm d}x\leq\epsilon, \limsup\limits_{n\rightarrow \infty}\int_{B^{c}_{r}}|v_{n}|^{q}{\rm d}x\leq\epsilon, \end{equation} $

(2.7) $ \begin{equation} \limsup\limits_{n\rightarrow \infty}\int_{B^{c}_{r}}F(x, u_{n}, v_{n}){\rm d}x\leq\epsilon, \int_{B^{c}_{r}}F(x, u, v){\rm d}x\leq\epsilon. \end{equation} $

因此, 有

(2.8) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}|u_{n}|^{q}{\rm d}x=\int_{{{\Bbb R}} ^{N}}|u|^{q}{\rm d}x, \lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}|v_{n}|^{q}{\rm d}x=\int_{{{\Bbb R}} ^{N}}|v|^{q}{\rm d}x, \end{equation} $

(2.9) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}F(x, u_{n}, v_{n}){\rm d}x=\int_{{{\Bbb R}} ^{N}}F(x, u, v){\rm d}x. \end{equation} $

引理2.3  假定条件$ \rm (H1)–(H2) $成立. 令$ \{(u_{n}, v_{n})\} $是泛函$ J $在$ X $中的$ (PS) $序列且满足(2.5)式, 则以下结论成立:

$ \rm (ⅰ) $对于每个足够小的$ \varepsilon>0, $存在$ r_{0}>1 $使得当$ r>r_{0} $时, 有

(2.10) $ \begin{equation} \int_{B^{c}_{2r}}(|\nabla u_{n}|^{p}+V(x)|u_{n}|^{p}){\rm d}x+\int_{B^{c}_{2r}}(|\nabla v_{n}|^{p}+V(x)|v_{n}|^{p}){\rm d}x\leq\varepsilon, \end{equation} $

(2.11) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}V(x)|u_{n}-u|^{p}{\rm d}x=0, \lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}V(x)|v_{n}-v|^{p}{\rm d}x=0; \end{equation} $

$ \rm (ⅱ) $$ (u, v)\in X $是泛函$ J $的一个临界点.

证  (ⅰ) 对任意的$ r>1 $, 令函数$ \eta_{r}=\eta_{r}(|x|)\in C^{1}({{\Bbb R}} ^{N}) $使得

(2.12) $ \begin{equation} \eta_{r}(|x|)\equiv1, x\in B^{c}_{2r}, \eta_{r}(|x|)=0, x\in B_{r}, 0\leq\eta_{r}\leq1. \end{equation} $

并且

(2.13) $ \begin{equation} 0\leq\eta_{r}\leq1, |\nabla\eta_{r}|\leq\frac{2}{r}, x\in{{\Bbb R}} ^{N}. \end{equation} $

由于$ \{(u_{n}, v_{n})\} $在$ X $中有界, 从而$ \{(\eta_{r}u_{n}, \eta_{r}v_{n})\} $在$ X $中有界. 因此当$ n\rightarrow \infty $时, 有

(2.14) $ \begin{equation} J'(u_{n}, v_{n})(\eta_{r}u_{n}, \eta_{r}v_{n})=o_{n}(1), \end{equation} $

其中

(2.15) $ \begin{eqnarray} J'(u_{n}, v_{n})(\eta_{r}u_{n}, \eta_{r}v_{n})&=&\|u_{n}\|^{pk}_{E}\int_{{{\Bbb R}} ^{N}}(|\nabla u_{n}|^{p} +V(x)|u_{n}|^{p})\eta_{r}{\rm d}x\\ &&+\|v_{n}\|^{pk}_{E}\int_{{{\Bbb R}} ^{N}}(|\nabla v_{n}|^{p} +V(x)|v_{n}|^{p})\eta_{r}{\rm d}x\\ &&+A_{n}(r)+B_{n}(r)+C_{n}(r)+D_{n}(r), \end{eqnarray} $

这里

(2.16) $ \begin{eqnarray} A_{n}(r)&=&-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}(F_{u_n}(x, u_{n}, v_{n})u_{n}+F_{v_n}(x, u_{n}, v_{n})v_{n})\eta_{r}{\rm d}x\\ &=&-\sigma\int_{{{\Bbb R}} ^{N}}F(x, u_{n}, v_{n})\eta_{r}{\rm d}x; \end{eqnarray} $

(2.17) $ \begin{eqnarray} B_{n}(r)&=&\|u_n\|_E ^{pk}\int_{{{\Bbb R}} ^{N}}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla\eta_{r}u_{n}{\rm d}x; \end{eqnarray} $

(2.18) $ \begin{eqnarray} C_{n}(r)&=&\|v_n\|_E ^{pk}\int_{{{\Bbb R}} ^{N}}|\nabla v_{n}|^{p-2}\nabla v_{n}\nabla\eta_{r}v_{n}{\rm d}x; \end{eqnarray} $

(2.19) $ \begin{eqnarray} D_{n}(r)&=&-\int_{{{\Bbb R}} ^{N}}(\lambda|u_{n}|^{q}+\mu|v_{n}|^{q}|)\eta_{r}{\rm d}x. \end{eqnarray} $

因此由(2.7)式可知, 当$ n $足够大时有

(2.20) $ \begin{equation} |A_{n}(r)|\leq|\sigma|\int_{ B^{c}_{r}}(F(x, u_{n}, v_{n})\eta_{r}{\rm d}x\rightarrow0, r\rightarrow \infty. \end{equation} $

设$ \Omega_{r}=B^{c}_{r}\setminus\overline{B^{c}_{2r}} $, 对任何$ n\in N $, 当$ r\rightarrow \infty $时, 有

(2.21) $ \begin{eqnarray} |B_{n}(r)|&\leq&\|u_{n}\|^{pk}_{E}\int_{\Omega_{r}}|\nabla u_{n}|^{p-1}|\nabla\eta_{r}||u_{n}|{\rm d}x \leq \frac{2}{r}\|u_{n}\|^{pk}_{E}\int_{ \Omega_{r}}|\nabla u_{n}|^{p-1}|u_{n}|{\rm d}x\\ &\leq&\frac{2}{r}\|u_{n}\|^{pk}_{E}\|\nabla u_{n}\|^{p-1}_{L^{p}(\Omega_{r})}\|u_{n}\|_{L^{p}(\Omega_{r})} \leq \frac{2}{r}\|u_{n}\|^{m}_{E}\leq\frac{2}{r}M_0 ^{m}\rightarrow0; \end{eqnarray} $

(2.22) $ \begin{eqnarray} |C_{n}(r)|&\leq&\|v_{n}\|^{pk}_{E}\int_{\Omega_{r}}|\nabla v_{n}|^{p-1}|\nabla\eta_{r}||v_{n}|{\rm d}x \leq \frac{2}{r}\|v_{n}\|^{pk}_{E}\int_{ \Omega_{r}}|\nabla v_{n}|^{p-1}|v_{n}|{\rm d}x\\ &\leq&\frac{2}{r}\|v_{n}\|^{pk}_{E}\|\nabla v_{n}\|^{p-1}_{L^{p}(\Omega_{r})}\|v_{n}\|_{L^{p}(\Omega_{r})} \leq\frac{2}{r}\|v_{n}\|^{m}_{E}\leq\frac{2}{r}M_0^{m}\rightarrow0; \end{eqnarray} $

(2.23) $ \begin{eqnarray} |D_{n}(r)|&\leq&\lambda\|u_{n}\|^{q}_{L^{q}(B^{c}_{r})}+\mu\|v_{n}\|^{q}_{L^{q}(B^{c}_{r})}\rightarrow0. \end{eqnarray} $

又由于$ \|u_n\|_E \leq M_0, \|v_n\|_E \leq M_0 $, 根据(2.14)–(2.23)式, 存在常数$ r_{0}>1 $, 使得当$ r>r_{0} $时(2.10)式成立. 从而当$ r>r_{0} $时, 有

(2.24) $ \begin{equation} \int_{B^{c}_{2r}}V(x)(|u_{n}|^{p}+|v_{n}|^{p}){\rm d}x\leq\varepsilon, \end{equation} $

由(2.5)和(2.24)式, 可得

(2.25) $ \begin{equation} \int_{B^{c}_{2r}}V(x)(|u|^{p}+|v|^{p}){\rm d}x\leq\varepsilon. \end{equation} $

由于在$ L^{p}(B_{2r}) $中$ u_{n}\rightarrow u, v_{n}\rightarrow v, $因此有

(2.26) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\int_{B_{2r}}V(x)|u_{n}|^{p}{\rm d}x=\int_{B_{2r}}V(x)|u|^{p}{\rm d}x, \\ &&\lim\limits_{n\rightarrow \infty}\int_{B_{2r}}V(x)|v_{n}|^{p}{\rm d}x=\int_{B_{2r}}V(x)|v|^{p}{\rm d}x. \end{eqnarray} $

根据(2.24)和(2.26)式可得

(2.27) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}V(x)|u_{n}|^{p}{\rm d}x=\int_{{{\Bbb R}} ^{N}}V(x)|u|^{p}{\rm d}x, \\ &&\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}V(x)|v_{n}|^{p}{\rm d}x=\int_{{{\Bbb R}} ^{N}}V(x)|v|^{p}{\rm d}x. \end{eqnarray} $

因此由Brezis-Lieb引理^[7]即得(2.11)式成立.

(ⅱ) 根据(2.5)式可知, 当$ n\rightarrow \infty $时有

(2.28) $ \begin{eqnarray} &&\int_{{{\Bbb R}} ^{N}}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla\varphi {\rm d}x\rightarrow \int_{{{\Bbb R}} ^{N}}|\nabla u|^{p-2}\nabla u\nabla\varphi {\rm d}x, \forall\varphi\in C^{\infty}_{0}({{\Bbb R}} ^{N}), \\ &&\int_{{{\Bbb R}} ^{N}}|\nabla v_{n}|^{p-2}\nabla v_{n}\nabla\psi {\rm d}x\rightarrow \int_{{{\Bbb R}} ^{N}}|\nabla v|^{p-2}\nabla v\nabla\psi {\rm d}x, \forall\psi\in C^{\infty}_{0}({{\Bbb R}} ^{N}). \end{eqnarray} $

由(ⅰ)的证明, 当$ n\rightarrow \infty $时, 有

(2.29) $ \begin{eqnarray} &&\int_{{{\Bbb R}} ^{N}}V(x)(|\nabla u_{n}|^{p-2}u_{n}-|\nabla u|^{p-2}u)\varphi {\rm d}x\rightarrow0, \forall\varphi\in C^{\infty}_{0}({{\Bbb R}} ^{N}), \\ &&\int_{{{\Bbb R}} ^{N}}V(x)(|\nabla v_{n}|^{p-2}v_{n}-|\nabla v|^{p-2}v)\psi {\rm d}x\rightarrow0, \forall\psi\in C^{\infty}_{0}({{\Bbb R}} ^{N}), \end{eqnarray} $

从而可得

$ \int_{{{\Bbb R}} ^{N}}(F_{u}(x, u_{n}, v_{n})\varphi+F_{v}(x, u_{n}, v_{n})\psi){\rm d}x\rightarrow \int_{{{\Bbb R}} ^{N}}(F_{u}(x, u, v)\varphi+F_{v}(x, u, v)\psi){\rm d}x, \nonumber $

以及

(2.30) $ \begin{equation} \int_{{{\Bbb R}} ^{N}}(\lambda|u_{n}|^{q-2}u_{n}\varphi+\mu|v_{n}|^{q-2}v_{n}\psi){\rm d}x\rightarrow \int_{{{\Bbb R}} ^{N}}(\lambda|u|^{q-2}u\varphi+\mu|v|^{q-2}v\psi){\rm d}x. \end{equation} $

从而, 由(2.28)–(2.30)式及在$ X^{*} $中$ J'(u_{n}, v_{n})\rightarrow 0(n\rightarrow0) $, 可得

$ \lim\limits_{n\rightarrow \infty}J'(u_{n}, v_{n})(\varphi, \psi)=J'(u, v)(\varphi, \psi)=0, \forall\varphi, \psi\in C^{\infty}_{0}({{\Bbb R}} ^{N}).\nonumber $

由于$ C^{\infty}_{0}({{\Bbb R}} ^{N})\times C^{\infty}_{0}({{\Bbb R}} ^{N}) $在$ X $中是稠密的, 从而$ J'(u, v)(\varphi, \psi)=0, \forall(\psi, \varphi)\in X $. 因此, $ (u, v) $是$ J $在$ X $中的一个临界点. 证毕.

引理2.4  假设$ \rm (H1)–(H2) $成立. 若$ \{(u_{n}, v_{n})\} $是泛函$ J $在$ X $中的$ (PS) $序列且满足(2.5) 式, 则在$ X $中有$ (u_{n}, v_{n})\rightarrow (u, v) $, 即泛函$ J $在$ X $中满足$ (PS) $条件.

证  由于在$ X^{*} $中$ J'(u_{n}, v_{n})\rightarrow 0, n\rightarrow \infty $且$ (u, v) $是$ J $的临界点, 从而

(2.31) $ \begin{eqnarray} &&o_{n}(1)=J'(u_{n}, v_{n})(u_{n}, 0)=\|u_{n}\|^{pk+1}_{E}-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F_{u}(x, u_{n}, v_{n})u_{n}{\rm d}x- \int_{{{\Bbb R}} ^{N}}\lambda|u_{n}|^{q}{\rm d}x, \end{eqnarray} $

(2.32) $ \begin{eqnarray} &&o_{n}(1)=J'(u_{n}, v_{n})(0, v_{n})=\|v_{n}\|^{pk+1}_{E}-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F_{v}(x, u_{n}, v_{n})v_{n}{\rm d}x- \int_{{{\Bbb R}} ^{N}}\mu|v_{n}|^{q}{\rm d}x, \end{eqnarray} $

对(2.31)式, (2.32)式两边取极限, 得

(2.33) $ \begin{eqnarray} &&J'(u, v)(u, 0)=\|u\|^{pk+1}_{E}-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F_{u}(x, u, v)u{\rm d}x- \int_{{{\Bbb R}} ^{N}}\lambda|u|^{q}{\rm d}x=0, \\ &&J'(u, v)(0, v)=\|v\|^{pk+1}_{E}-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F_{v}(x, u, v)v{\rm d}x- \int_{{{\Bbb R}} ^{N}}\mu|v|^{q}{\rm d}x=0. \end{eqnarray} $

由(1.2)式和引理2.3的证明可知

(2.34) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}(F_{u}(x, u_{n}, v_{n})u_{n}-F_{u}(x, u, v)u){\rm d}x=0, \\ &&\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}(F_{v}(x, u_{n}, v_{n})v_{n}-F_{v}(x, u, v)v){\rm d}x=0. \end{eqnarray} $

从而, 根据(2.8)式, (2.27)式及(2.32)–(2.34)式, 可得

(2.35) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}|\nabla u_{n}|^{p}{\rm d}x=\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}|\nabla u|^{p}{\rm d}x, \end{eqnarray} $

(2.36) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}|\nabla v_{n}|^{p}{\rm d}x=\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^{N}}|\nabla v|^{p}{\rm d}x. \end{eqnarray} $

对(2.35)式, (2.36)式应用于Brezis-Lieb引理可得

(2.37) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\|\nabla(u_{n}-u)\|^{p}_{p}=0, \lim\limits_{n\rightarrow \infty}\|\nabla(v_{n}-v)\|^{p}_{p}=0, \end{equation} $

因此, 根据(2.11)式和(2.37)式可知序列$ {(u_{n}, v_{n})} $在空间$ X $中收敛. 从而, 泛函$ J $在$ X $满足$ (PS) $条件. 证毕.

引理3.1^[4]  设$ X $是无穷维实Banach空间, 泛函$ I\in C^{1}(X, {{\Bbb R}} ) $满足$ (PS) $条件的偶泛函, 且有$ I(0)=0 $. 若$ X=U\oplus V $ (其中$ U $是有限维), 且泛函$ I $满足

$ \rm (ⅰ) $存在常数$ \rho, \alpha>0 $, 使得当$ z\in \partial B_{\rho}\cap V $时, 有$ I(z)\geq\alpha; $

$ \rm (ⅱ) $对于空间$ X $的任意一个有限维子空间$ X_{0}\subset X, $存在正常数$ R=R(X_{0}), $使得任意的$ z\in X_{0}\setminus B_{R} $, 有$ I(z)\leq 0 $, 其中

$ B_{R}=\{z\in X:\|z\|<R\}, \partial B_{R}=\{z\in X:\|z\|=R\}. $

则$ I $有无穷多个临界点.

本文的主要结论为:

定理3.1  如果条件(H1)–(H2) 成立, 则问题(1.1)存在无穷多个非负弱解$ (u_{n}, v_{n})\in X $且当$ n\to \infty $时$ J(u_n, v_n)\to \infty $.

证  由泛函$ J $的表达式(2.3)式知$ J $是$ X $上的偶泛函, 且有$ J(0, 0)=0. $由引理2.4知泛函$ J $满足$ (PS) $条件. 接下来将我们证明如果$ \rm (H1)–(H2) $成立, 则泛函$ J $满足引理3.1的条件(ⅰ)和(ⅱ).

由(1.2)式和Sobolev嵌入不等式(2.1)和(2.2)可得

(3.1) $ \begin{eqnarray} J(u, v)&=&\frac{1}{m}(\|u\|^{m}_{E}+\|v\|^{m}_{E})-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F(x, u, v){\rm d}x -\frac{1}{q}\int_{{{\Bbb R}} ^{N}}(\lambda|u|^{q}+\mu|v|^{q}){\rm d}x\\ &\geq&\frac{1}{m}(\|u\|^{m}_{E}+\|v\|^{m}_{E})-\frac{c_{0}\sigma}{d}\int_{{{\Bbb R}} ^{N}}(|u|^{d}+|v|^{d}){\rm d}x -\frac{1}{q}\int_{{{\Bbb R}} ^{N}}(\lambda|u|^{q}+\mu|v|^{q}){\rm d}x\\ &\geq&\frac{1}{m}\|(u, v)\|^{m}-C_{1}(\|(u, v)\|^{d}+\|(u, v)\|^{q})\\ &\geq &m^{-1}\rho^{m}-2C_{1}\rho^{q}\geq m^{-1}\rho^{m}\equiv\alpha>0, \end{eqnarray} $

其中$ C_{1}>0 $且$ \|(u, v)\|=\rho=\{1, (2mC_{1})^{\frac{1}{m-t}}\}. $因此, 泛函$ J $满足引理3.1的条件(ⅰ).

接下来我们证明泛函$ J $满足引理3.1的条件(ⅱ).

设$ X_{0} $是空间$ X $的任意有限维子空间, 则$ X_{0} $上的所有范数等价. 因此, 存在常数$ \gamma>0 $使得

(3.2) $ \begin{equation} \|u\|_{q}+\|u\|_{q}\geq\gamma\|(u, v)\|, \forall(u, v)\in X_{0}. \end{equation} $

从而, 当$ \sigma\geq0 $时, 有

(3.3) $ \begin{eqnarray} J(u, v)&=&\frac{1}{m}(\|u\|^{m}_{E}+\|v\|^{m}_{E})-\frac{\sigma}{d}\int_{{{\Bbb R}} ^{N}}F(x, u, v){\rm d}x -\frac{1}{q}\int_{{{\Bbb R}} ^{N}}(\lambda|u|^{q}+\mu|v|^{q}){\rm d}x\\ &\leq&\frac{1}{m}(\|u\|^{m}_{E}+\|v\|^{m}_{E})-\frac{\lambda_{0}}{q}\gamma^{q}2^{1-q}\|(u, v)\|^{q}, \\ &\leq&\frac{1}{m}\|(u, v)\|^{m}-\frac{\lambda_{0}}{q}\gamma^{q}2^{1-q}\|(u, v)\|^{q}, \end{eqnarray} $

其中$ \lambda_{0}=\min\{\lambda, \mu\}>0. $由于$ m<q, $从而由(3.3)式知: 存在$ R_{0}>\rho, $使得当$ R>R_{0}, $$ \|(u, v)\|\geq R $时, 有$ J(u, v)<0 $. 即泛函$ J $满足引理3.1的条件(ⅱ).

由引理3.1可得: 问题$ (1.1) $存在无穷多个解$ (u_n, v_n)\in X $且当$ n\rightarrow \infty $时, $ J(u_{n}, v_{n})\rightarrow \infty. $因为$ J(u_{n}, v_{n})=J(|u_{n}|, |v_{n}|) $, 从而$ J'(-u_{n}, -v_{n})=-J'(u_{n}, v_{n}) $, 所以可以认为$ u_{n}, v_{n} $均为非负函数. 证毕.