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2022 Volume 47 Issue 8
Article Contents

SUN Xin, DUAN Yu. Existence and Multiplicity of Solutions for a Class of Klein-Gordon-Maxwell Systems[J]. Journal of Southwest China Normal University(Natural Science Edition), 2022, 47(8): 38-47. doi: 10.13718/j.cnki.xsxb.2022.08.006
Citation: SUN Xin, DUAN Yu. Existence and Multiplicity of Solutions for a Class of Klein-Gordon-Maxwell Systems[J]. Journal of Southwest China Normal University(Natural Science Edition), 2022, 47(8): 38-47. doi: 10.13718/j.cnki.xsxb.2022.08.006

Existence and Multiplicity of Solutions for a Class of Klein-Gordon-Maxwell Systems

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  • Corresponding author: DUAN Yu
  • Received Date: 19/12/2021
    Available Online: 20/08/2022
  • MSC: O177.91

  • In this paper, the existence and multiplicity of solutions have been established for a class of Klein-Gordon-Maxwell system with parameters and concave-convex nonlinearities. When the convex nonlinearity is general superlinear, the existence and multiplicity of solutions for the system have been proved via variational methods and make accurate analysis on the combined effect of parameters. Our result completes some recent works concerning the existence of solutions of this system.
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Existence and Multiplicity of Solutions for a Class of Klein-Gordon-Maxwell Systems

    Corresponding author: DUAN Yu

Abstract: In this paper, the existence and multiplicity of solutions have been established for a class of Klein-Gordon-Maxwell system with parameters and concave-convex nonlinearities. When the convex nonlinearity is general superlinear, the existence and multiplicity of solutions for the system have been proved via variational methods and make accurate analysis on the combined effect of parameters. Our result completes some recent works concerning the existence of solutions of this system.

  • 研究如下Klein-Gordon-Maxwell系统:

    其中: $\omega>0$是一个常数; $1<s<2, \lambda, \mu$是参数; $u, \phi:$ $\mathbb{R}^{3} \longrightarrow \mathbb{R}, f, g:$ $\mathbb{R}^{3} \times \mathbb{R} \longrightarrow$皿. 系统(1) 起源于数学物理领域中的某些应用问题. 为了描述三维空间中非线性Klein-Gordon场与静电场之间相互作用所产生的孤立波问题, 文献[1] 首次提出了如下Klein-Gordon-Maxwell系统模型:

    其中0 < ω < m0,4 < q < 6,m0e分别表示粒子的质量和电量,而ω表示相位.系统的未知因素是联系粒子的场u和电磁位势ϕ. 有关系统(2)物理方面的详述可参见文献[1-2]. 作为系统(2)的一般情形,系统(1)近年来受到了众多学者的关注(见文献[3-25]). 需特别提及的是:文献[15-16]考虑了系统(1)无穷多解的存在性问题,但其所考虑的凹凸非线性项均是特殊的非线性项,且凸非线性项满足超4次条件;文献[22]在非线性项满足局部(AR)条件时考虑了系统(1)解的存在性和多重性.

    本文的主要目的是当凸非线性项是不满足局部(AR)条件的一般非线性项,且非线性项含有两个参数时,利用变分法讨论两个参数对系统(1)解的存在性和多重性的具体影响. 本文的结论完善了已有文献的相关结果.

    本文针对位势函数V及非线性项αf做如下假设:

    (V) $V \in C { (\mathbb{R}^{3}, \mathbb{R}), } $ $\inf\limits _{x \in \mathbb{R}^{3}} V(x)>0 \text {, 且对 } \forall M>0 \text {, 有 } \operatorname{meas}\left\{x \in \mathbb{R}^{3}: V(x) \leqslant M\right\}<\infty$;

    $\left(\mathrm{F}_{1}\right) f \in C\left(\mathbb{R}^{3} \times \mathbb{R}, \mathbb{R}\right.$), $\lim _{t \rightarrow 0} \frac{f(x, t)}{t}=0$关于$x \in \mathbb{R}^{3}$一致成立;

    $\left(\mathrm{F}_{2}\right)$存在常数$C_{1}>0$$2<p<6$, 满足$|f(x, t)| \leqslant C_{1}\left(1+|u|^{p-1}\right), \forall(x, t) \in$ $\mathbb{R}^{3} \times \mathbb{R}^{*}$;

    $\left(\mathrm{F}_{3}\right)$存在常数$\theta>0, r_{0}>0$, 使得$\widetilde{F}(x, t)=f(x, t) t-2 F(x, t) \geqslant \theta|t|^{p}, \forall x \in \mathbb{R}^{3}, |u| \geqslant r_{0}$;

    $\left(\mathrm{F}_{4}\right) \lim _{|t| \rightarrow \infty} \frac{F(x, t)}{|t|^{2}}=+\infty$关于$x \in$ $\mathbb{R}^{3}$一致成立, 且$f(x, t) t \geqslant 2 F(x, t) \geqslant 0, \forall(x, t) \in$ $\mathbb{R}^{3} \times \boldsymbol{R}^{\prime} ;$

    $\left(\mathrm{F}_{5}\right) f(x, -t)=-f(x, t), \forall(x, t) \in \mathbb{R}^{3} \times \mathbb{R}$;

    (A) $\alpha(x) \in L^{\frac{2}{2-s}}\left(\mathbb{R}^{3}\right), 1<s<2, \alpha(x) \geqslant 0$.

    定理1    假设Vfα分别满足条件(V),(F1)-(F4)及(A),则:

    (i) 对$\forall \lambda>0$, 存在$\bar{\mu}_{\lambda}>0$, 使得对$\forall \mu \in\left(-\bar{\mu}_{\lambda}, \bar{\mu}_{\lambda}\right.$), 系统(1) 至少有一个非平凡解;

    (ii) 对$\forall \lambda>0$, 存在$\bar{\mu}_{\lambda}>0$, 使得对$\forall \mu \in\left(0, \bar{\mu}_{\lambda}\right)$, 系统(1) 至少有两个非平凡解;

    (iii) 对$\forall \mu>0$, 存在$\bar{\lambda}_{\mu}>0$, 使得对$\forall \lambda \in\left(0, \bar{\lambda}_{\mu}\right)$, 系统(1) 至少有两个非平凡解.

    定理2    假设Vfα分别满足条件(V),(F1)-(F5)及(A),则对$\forall \lambda>0, \mu \in \mathbb{R}$,系统(1)有一列高能量解.

    定理3    假设Vfα分别满足条件(V),(F1)-(F5)及(A),则对$\forall \lambda>0, \mu>0$,系统(1)有一列负能量解.

    注1    确实存在函数满足条件(F1)-(F4),但不满足(AR)条件及超4次条件,如

    注2    与文献[15-16]的结论相比,本文在凸非线性项是一般非线性项,且不满足超4次条件下讨论了系统(1)解的存在性和多重性.

    与文献[22]的结论相比,本文从两个方面改进了其结果:

    (i) 本文在非线性项不满足局部(AR)条件而满足更弱的超线性条件时给出了系统(1)解的存在性和多重性结果;

    (ii) 本文在参数μ满足更大范围的限制性条件下仍获得了与其相同的结果:系统(1)有一个非平凡解和一列高能量解,且本文还讨论了一列负能量解的存在性问题.

    因此本文完善了已有文献的相关结果.

    表示Sobolev空间,其范数定义为

    $H^{1}\left(\mathbb{R}^{3}\right)=\left\{u \in L^{2}\left(\mathbb{R}^{3}\right): \nabla u \in L^{2}\left(\mathbb{R}^{3}\right)\right\}$,其内积和范数分别定义为

    定义

    由条件(V),H是Hilbert空间,其内积和范数分别定义为

    显然,在条件(V)下,对任意的2≤p < 6,嵌入映射$H \circlearrowleft L^{p}\left(\mathbb{R}^{3}\right)$是紧映射,且对任意的2≤p≤6,存在Sp>0,使得

    系统(1)具有变分结构,对$\forall(u, \phi) \in H \times D^{1, 2}\left(\mathbb{R}^{3}\right)$,定义其能量泛函

    易知系统(1) 的弱解$(u, \phi) \in H \times D^{1, 2}(\mathbb{R}^{3})$对应着泛函$J$的临界点. 由于$J$是强不定的, 为了克服这种困难, 需要对泛函进行一些简化, 将泛函$J$转化成只含有一个变量$u$的式子.

    引理1[3]    对$\forall u \in H^{1}\left(\mathbb{R}^{3}\right) \text {, 存在唯一的 } \phi=\phi_{u} \in D^{1, 2}\left(\mathbb{R}^{3}\right)$,满足

    更进一步,映射$\mathit{\Phi}: u \in H^{1}\left(\mathbb{R}^{3}\right) \longrightarrow \mathit{\Phi}[u]=\phi_{u} \in D^{1, 2}\left(\mathbb{R}^{3}\right)$是连续可微的,并且满足:

    (i) 在集合$\{x: u(x) \neq 0\}$上, $-\omega \leqslant \phi_{u} \leqslant 0$;

    (ii) $\left\|\phi_{u}\right\|_{D 1, 2} \leqslant C\|u\|^{2}$, 且$\int_{\mathbb{R}^{3}}\left|\phi_{u}\right| u^{2} \mathrm{~d} x \leqslant C\|u\|^{4}$.

    在(4)式左右两端同时乘以ϕu,并分部积分,可得

    结合(5)式及J的定义知,I(u)=J(uϕu)可化简为

    由条件(V),(F1)-(F3)及引理1易知,I定义在空间H上是有意义的,且$I \in C^{1}(H, \mathbb{R})$,其所对应的导数为

    由文献[1]的命题$3.5$知, $u$是泛函$I$的临界点当且仅当$(u, \phi) \in H \times D^{1, 2}\left(\mathbb{R}^{3}\right)$是系统(1) 的解, 并且$\phi=\phi_{u}$. 因此, 为了得到系统(1) 的非零解, 我们只需寻找泛函$I$的非零临界点即可.

    $\left\{e_{i}\right\}$为空间$H$的一组正交基. $X_{i}=\mathbb{R}e_{i}, Y_{k}=\bigoplus_{i=1}^{k} X_{i}, Z_{k}=\oplus_{i=k+1}^{\infty} X_{i}, k \in \mathbb{N}$. 为了证明定理$1-$定理3, 我们给出以下几个引理:

    引理2[26]    设$1<s<2<r, A, B>0$, 令$\mathit{\Psi}_{A, B}(t)=t^{2}-A t^{s}-B t^{r}(t \geqslant 0)$. 则$\max\limits _{t \geqslant 0} \mathit{\Psi}_{A, B}(t)>0$当且仅当$A^{r-2} B^{2-s}<d(r, s)=\frac{(r-2)^{r-2}(2-s)^{2-s}}{(r-s)^{r-s}}$. 若$t=t_{B}=\left[\frac{2-s}{B(r-s)}\right]^{\frac{1}{r-2}}$, 则有

    引理3    假设条件(V),(F1)-(F4)及(A)成立,则对$\forall \lambda>0, \mu \in \mathbb { R }$I在空间H满足(PS)c条件.

        设$ \left\{u_{n}\right\} \subset H$是泛函I的任一(PS)c序列,即

    从而存在常数M>0,使得

    首先证明:(PS)c序列{un}有界.

    下面采用反证法证明$\left\|u_{n}\right\|$有界. 假设存在$\left\{u_{n}\right\}$的一个子列(不失一般性, 仍记此子列为$\left\{u_{n}\right\}$), 使得$\left\|u_{n}\right\| \rightarrow \infty$. 令$\omega_{n}=\frac{u_{n}}{\left\|u_{n}\right\|}$, 则$\left\|\omega_{n}\right\|=1$. 因为对任意的$2 \leqslant p<6$, 嵌人映射$H \cup L^{p}\left(\mathbb{R}^{3}\right)$是紧的, 所以存在$\left\{\omega_{n}\right\}$的一个子列(不失一般性, 仍记之为$\left\{\omega_{n}\right\}$) 和$\omega_{0} \in H$, 使得

    $\mathit{\Omega}=\left\{y \in \mathbb{R}^{3}: \omega_{0}(y) \neq 0\right\}$. 若meas$(\mathit{\Omega})>0$, 则$\left|u_{n}\right|=\left|\omega_{n}\right|\left\|u_{n}\right\| \rightarrow \infty$ (a. e. $x \in \mathit{\Omega}(n \rightarrow \infty)$).

    由条件(F4)和Fatou引理知

    而由引理1(i)及(6)式知

    这显然与(7) 式是矛盾的, 故$\operatorname{meas}(\mathit{\Omega})=0$. 从而有$\omega_{0}=0$, 且在$L^{p}\left(\mathbb{R}^{3}\right)(2 \leqslant p<6)$$\omega_{n} \rightarrow 0$.

    由条件(F3)-(F4)及引理1(i)知

    因在$L^{p}\left(\right.$ R $\left.^{3}\right)(2 \leqslant p<6)$$\omega_{n} \rightarrow 0$, 则

    由条件(F1)-(F2)知,对$ \forall \varepsilon>0$,存在Cε>0,满足

    故由条件(F1)-(F3),(A),(6),(8),(9)式及引理1(i)知

    这显然是矛盾的,故序列{un}是有界的.

    其次证明{un}在空间H中有一个强收敛的子列.

    由文献[14]中引理2.4的证明过程知

    故由$(10)-(15)$式知, 在空间$H$$u_{n} \rightarrow u(n \rightarrow \infty)$.

    引理4    假设条件(V),(F1)-(F4)及(A)成立,则泛函I(u)满足如下山路结构:

    (i) 对$\forall \lambda>0$, 存在$\bar{\mu}_{\lambda}>0, \alpha>0, \rho>0$, 使得对$\forall \mu \in\left(-\bar{\mu}_{\lambda}, \bar{\mu}_{\lambda}\right), I(u) \mid\|u\|=\rho \geqslant \alpha>0$, 或对$\forall \mu>0$, 存在$\bar{\lambda}_{\mu}>0, \alpha>0, \rho>0$, 使得对$\forall \lambda \in\left(0, \bar{\lambda}_{\mu}\right), \left.I(u)\right|_{\|u\|=\rho} \geqslant \alpha>0$;

    (ii) 对$\forall \lambda>0, \mu \in$目, 存在$e \in H$满足$\|e\|>\rho$, 使得$I(e)<0$.

        (i) 由(9)式及引理1(i)知

    $\varepsilon \in\left(0, \frac{1}{2 S_{2}^{2} \lambda}\right]$,则

    $D=\frac{4}{s}\|\alpha\| \frac{2}{2-s} S_{2}^{s}, E=\frac{4 C_{\varepsilon}}{p} S_{p}^{p}$, 取$r=p$, 则由引理2知, 当$A^{p-2} B^{2-s}<d(p, s)$时, 有

    其中

    即对$\forall \lambda>0$, 当$|\mu|<\frac{1}{D}\left(d(p, s)(\lambda E)^{s-2}\right)^{\frac{1}{p-2}}=\bar{\mu}_{\lambda}$, 或对$\forall \mu>0$, 当$\lambda<\frac{1}{E}\left(d(p, s)(\mu D)^{2-p}\right)^{\frac{1}{2-s}}=\bar{\lambda}_{\mu}$时, $I(u) \geqslant \frac{1}{4} \mathit{\Psi}_{A, B}\left(t_{B}\right)>0$. 令$\alpha=\frac{1}{4} \mathit{\Psi}_{A, B}\left(t_{B}\right), \rho=t_{B}$, 则$\alpha>0$$\left.I\right|_{\|u\|=\rho} \geqslant \alpha>0$.

    (ii) 因为

    所以对$\forall t>0, u \in H \backslash\{0\}$, 有

    故由条件(F4)及Fatou引理知

    因此存在$t_{0}>0, e=t_{0} u$, 满足$\|e\|>\rho$, 使得$I(e)<0$.

    引理5    假设条件(V),(F1)-(F4)及(A)成立,则对$\forall \lambda>0, \mu \in \mathbb{R}$,有:

    (i) 存在$\alpha>0, \rho>0$, 使得$\left.I\right|_{\partial B_{\rho} \cap Z_{k}} \geqslant \alpha$;

    (ii)对任意的有限维子空间$\widetilde{E} \subset H$, 存在$R=R(\widetilde{E})>0$, 使得$\left.I\right|_{\tilde{E} \backslash B_{R}} \leqslant 0$.

        (i) 在(9)式中取ε=ε0>0为某一给定的常数,则存在Cε0>0,满足

    $\beta_{k}=\mathop {{\rm{sup}}}\limits_{u \in {Z_k},\left\| u \right\| = 1} \| u \|_{p}, 2 \leqslant p<6$, 则$\beta_{k} \rightarrow 0, k \rightarrow \infty$. 从而可知, 存在$k_{1}>1$, 使得当$k>k_{1}$时有

    因为1 < s < 2,所以存在R0>0,使得当‖u‖≥R0

    由(16)-(18) 式及引理1(i) 知, $\forall u \in Z_{k},\|u\| \geqslant R_{0}$, 有

    $\rho=\left(4 \beta_{k}^{p} \lambda C_{\varepsilon_{0}}\right)^{\frac{1}{2-\rho}}, \alpha=\left(\frac{1}{8}-\frac{1}{4 p}\right) \rho^{2}$, 则$\rho>0, \alpha>0$$\rho \rightarrow \infty, k \rightarrow \infty$. 从而存在$k_{2}>1$, 使得当$k>k_{2}$时, $\rho>R_{0}$. 故当$k>\max \left\{k_{1}, k_{2}\right\}, u \in Z_{k}, \|u\|=\rho$时, $I(u) \geqslant \alpha=\left(\frac{1}{8}-\frac{1}{4 p}\right) \rho^{2}>0$.

    (ii) 设$\widetilde{E} \subset H$是任一有限维子空间. 现利用反证法证明. 假设存在序列$\left\{u_{n}\right\} \subset \widetilde{E}$满足$\left\|u_{n}\right\| \rightarrow+\infty$, 但$I\left(u_{n}\right)>0$. 令$v_{n}=\frac{u_{n}}{\left\|u_{n}\right\|}$, 则$\left\|v_{n}\right\|=1$. 因为$\widetilde{E} \subset H$是有限维子空间, 所以存在$\left\{v_{n}\right\}$的一个子列(不失一般性, 仍记之为$\left.\left\{v_{n}\right\}\right)$$v_{0} \in \widetilde{E}$, 使得$v_{n} \rightarrow v_{0}(x \in \widetilde{E}), \left\|v_{0}\right\|=1$. 故由条件($\left.\mathrm{F}_{4}\right)$及Fatou引理知

    这显然是矛盾的, 故存在$R=R(\widetilde{E})>0$, 使得$\left.I\right|_{\tilde{E} \backslash B_{R}} \leqslant 0$.

    引理6    假设条件$(\mathrm{V}), \left(\mathrm{F}_{1}\right)-\left(\mathrm{F}_{4}\right)$及(A) 成立, 则对$\forall \lambda>0, \mu>0$, 存在$k_{0} \in \mathbb{N}$, 使得对每个$k>k_{0}$, 存在$\rho_{k}>\gamma_{k}>0$且满足:

    (i) $a_{k}:=\inf\limits _{u \in Z_{k}, \|u\|=\rho_{k}} I(u) \geqslant 0$;

    (ii) $b_{k}:=\max\limits _{u \in Y_{k}, \|u\|=\gamma_{k}} I(u)<0$;

    (iii) $d_{k}:=\inf\limits _{u \in Z_{k}, \|u\| \leqslant \leqslant_{k}} I(u) \rightarrow 0, k \rightarrow+\infty$.

        (i)令$\beta_{k}=\sup\limits _{u \in Z_{k}, \left\|_{u}\right\|=1}\|u\|_{p}, 2 \leqslant p<6$, 则$\beta_{k} \rightarrow 0, k \rightarrow \infty$. 因为$2<p<6$, 所以存在$R_{1}>0$, 使得

    由(3),(9),(19)式及引理1(i)知,对$\forall u \in Z_{k}$, 有$\|u\| \leqslant R_{1}$

    $\varepsilon \in\left(0, \frac{1}{4 S_{2}^{2} \lambda}\right]$, 则

    $\rho_{k}=\left(8 \beta_{k}^{\xi} \mu\|\alpha\| \frac{2}{2-s} \frac{1}{2-s}\right.$, 则$\rho_{k} \rightarrow 0(k \rightarrow+\infty)$. 从而存在$k_{0} \in \mathbb{N}$, 使得当$k>k_{0}$时, $\rho_{k}<R_{1}$. 故当$k>k_{0}$, $u \in Z_{k}, \|u\|=\rho_{k}$时,

    即(i)成立.

    (ii) 对$\forall u \in Y_{k}, \delta>0$, 令$\mathit{\Gamma}_{\alpha, \delta}(u)=\left\{x \in \mathbb{R}^{3}: \alpha(x)|u|^{s} \geqslant \delta\|u\|{ }^{s}\right\}$, 由文献[4]中定理$1.5$的证明过程可知, 存在$\varepsilon_{1}>0$使得$\operatorname{meas}\left(\mathit{\Gamma}_{a, \varepsilon_{1}}(u)\right) \geqslant \varepsilon_{1}$.

    故结合条件$\left(\mathrm{F}_{4}\right), (\mathrm{A})$, (9) 式及引理$1(\mathrm{i})$知, 对$\forall u \in Y_{k}$, 有

    因为$1<s<2$, 所以存在$\gamma_{k} \in\left(0, \rho_{k}\right)$, 使得当$u \in Y_{k}, \|u\|=\gamma_{k}$时, $I(u) \leqslant 0$, 即(ii) 成立.

    (iii) 由(20) 式, 对$\forall u \in Z_{k}, \|u\| \leqslant \rho_{k}$, 有

    因为$\rho_{k} \rightarrow 0(k \rightarrow+\infty)$, 所以$\inf\limits _{u \in Z_{k}, \|u\| \leqslant \rho_{k}} I(u) \rightarrow 0(k \rightarrow+\infty)$, 即(iii) 成立.

    定理1的证明    (i) 证明系统(1) 存在一个山路解. 由引理4知, 对$\forall \lambda>0$, 存在$\bar{\mu}_{\lambda}>0$, 使得对$\forall \mu \in\left(-\bar{\mu}_{\lambda}, \bar{\mu}_{\lambda}\right)$, 泛函$I$满足山路定理的几何结构. 由引理3知, 泛函$I$满足(PS) 条件. 因此由山路定理(见文献[27] 的定理2.2) 知, 存在$u_{0} \in H$满足$I^{\prime}\left(u_{0}\right)=0$$I\left(u_{0}\right)>0$. 即系统(1) 存在一个山路解.

    (ii) 首先证明系统(1) 存在一个山路解. 由引理4知, 对$\forall \lambda>0$, 存在$\bar{\mu}_{\lambda}>0$, 使得对$\forall \mu \in\left(0, \bar{\mu}_{\lambda}\right)$, 泛函$I$满足山路定理的几何结构. 由引理3知, 泛函$I$满足(PS) c条件. 因此由山路定理(见文献[27]的定理2.2) 知, 存在$u_{0}^{\prime} \in H$满足$I^{\prime}\left(u_{0}^{\prime}\right)=0$$I\left(u_{0}^{\prime}\right)>0$. 即系统(1) 存在一个山路解.

    其次证明系统(1) 存在一个局部极小解. 由条件(A) 知, 存在$v \in H$使得$\int_{\mathbb{R}^{3}} a(x)|v|^{s} \mathrm{~d} x>0$. 结合条件$\left(\mathrm{F}_{4}\right)$及引理1 (i) 知, 当$\mu>0, t>0$充分小时,

    因此$c_{0}=\inf \left\{I(u): u \in \bar{B}_{\rho}\right\}<0$, 其中$B_{\rho}=\{u \in H:\|u\|<\rho\}, \rho$已由引理4给出. 利用Ekeland变分原理知, 存在一个有界的极小化序列$\left\{u_{n}\right\} \subset \bar{B}_{\rho}$, 满足$I\left(u_{n}\right) \rightarrow c_{0}, I^{\prime}\left(u_{n}\right) \rightarrow 0(n \rightarrow \infty)$. 故利用引理3知存在$u_{1} \in H$满足$I^{\prime}\left(u_{1}\right)=0, I\left(u_{1}\right)=c_{0}<0$.

    (iii) 首先证明系统(1) 存在一个山路解. 由引理4知, 对$\forall \mu>0$, 存在$\bar{\lambda}_{\mu}>0$, 使得对$\forall \lambda \in\left(0, \bar{\lambda}_{\mu}\right)$, 泛函$I$满足山路定理的几何结构. 由引理3知, 泛函$I$满足$(\mathrm{PS})_{c}$条件. 因此由山路定理知, 存在$u_{0}^{\prime \prime} \in H$满足$I^{\prime}\left(u_{0}^{\prime \prime}\right)=0$$I\left(u_{0}^{\prime \prime}\right)>0$. 即系统(1) 存在一个山路解.

    其次证明系统(1) 存在一个局部极小解. 证明同(ii), 略去.

    定理2的证明    由条件$\left(\mathrm{F}_{5}\right)$知泛函$I$是偶的. 结合引理3及引理5知, 能量泛函$I$满足对称山路定理(见文献[27]的定理9.12) 的条件. 因此$I$有一列趋于$+\infty$的临界值, 即系统(1) 具有一列高能量解.

    定理3的证明    由条件$\left(\mathrm{F}_{5}\right)$知泛函$I$是偶的. 结合引理3及引理6知, 能量泛函$I$满足对偶喷泉定理(见文献[28] 的定理3.18) 的条件. 故$I$有一列趋于0的负的临界值, 即系统(1) 存在一列负能量解.

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