西南大学学报 (自然科学版)  2019, Vol. 41 Issue (2): 64-69.  DOI: 10.13718/j.cnki.xdzk.2019.02.010
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  • 一类带有周期位势的分数阶耦合系统的正基态解    [PDF全文]
    贺书文, 商彦英     
    西南大学 数学与统计学院, 重庆 400715
    摘要:运用Ekeland变分原理研究了一类带有不同周期位势的分数阶耦合系统非平凡解的存在性.证明了该系统存在的非平凡解可以是一个正基态解,该结果将一般的Schrödinger耦合系统推广到带有多个不同周期函数的分数阶耦合系统的情形.
    关键词分数阶耦合系统    Ekeland变分原理    正基态解    

    考虑下面的分数阶耦合系统:

    $ \left\{ \begin{gathered} {\left( { - \Delta } \right)^s}u + {V_1}\left( x \right)u = {\lambda _1}\left( x \right){u^{p - 1}} + \beta {v^{\frac{p}{2}}}{u^{\frac{p}{2} - 1}}\;\;\;\;\;\;\;x \in {\mathbb{R}^N} \hfill \\ {\left( { - \Delta } \right)^s}v + {V_2}\left( x \right)v = {\lambda _2}\left( x \right){v^{p - 1}} + \beta {u^{\frac{p}{2}}}{v^{\frac{p}{2} - 1}}\;\;\;\;\;\;\;x \in {\mathbb{R}^N} \hfill \\ u,v > 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in {\mathbb{R}^N} \hfill \\ \end{gathered} \right. $ (1)

    其中:s∈(0,1),2sN;(-Δ)s为分数阶Laplace算子,且p∈(2,2s*),2s*=$ \frac{2N}{N-2s}$;对于i=1,2,Vi(x),λi(x)为正的有界连续函数;β>0为耦合常数.这类系统广泛地出现在非线性光学、材料学和凝聚理论等领域中.当s=1,β=0时,系统(1)便退化成两个一般的Schrödinger方程,这类问题已经被广泛研究(参见文献[1-3]等).当s=1,p=4,Vi(x)和λi(x)为不同的正常数时,系统(1)便转化为一般的非线性Schrödinger耦合系统,其非平凡正解的存在性结果已经有很多了(参见文献[4-7]等),特别是在文献[6-7]中,作者证明了对于一些常数β2β1>0,当β1β>0或ββ2时,该Schrödinger系统存在非平凡正解.当β=0时,在不同的假设条件下,对于系统(1)中单个方程的非平凡解的存在性也被考虑(参见文献[8-11]等).近年来,文献[12]考虑了下面的耦合系统:

    $ \left\{ \begin{gathered} {\left( { - \Delta } \right)^s}u + u = \left( {{{\left| u \right|}^{2p}} + b{{\left| u \right|}^{p - 1}}{{\left| v \right|}^{p + 1}}} \right)u\;\;\;\;\;\;\;\;\;\;\;x \in {\mathbb{R}^N} \hfill \\ {\left( { - \Delta } \right)^s}v + {\omega ^{2s}}v = \left( {{{\left| v \right|}^{2p}} + b{{\left| v \right|}^{p - 1}}{{\left| u \right|}^{p + 1}}} \right)v\;\;\;\;\;\;\;x \in {\mathbb{R}^N} \hfill \\ \end{gathered} \right. $ (2)

    在一定条件下,证明了系统(2)具有一系列低能量解的存在性结果.受文献[9-10, 12]的启发,本文利用(Ce)条件替代(PS)条件的山路定理、Nehari流形方法和集中紧性原理来研究系统(1)正基态解的存在性,主要困难在于解决缺失紧性和排除半平凡解(u,0)和(0,v).本文明显对文献[12]做了一定的推广,考虑带有多个不同周期函数的分数阶耦合系统正基态解的存在性,现陈述主要结果如下:

    定理1  假设下列条件成立:

    (H1) 2sNs ∈(0,1),p∈(2,2s*),2s*=$ \frac{2N}{N-2s}$,且β>0;

    (H2) Vi(x),λi(x) ∈C($ {{\mathbb{R}}^{N}}, \mathbb{R}$)∩L($ {{\mathbb{R}}^{N}}$)在每个x1x2,…,xN上是1-周期函数;

    (H3)存在常数Viλi>0,使得Vi(x)≥Viλi(x)≥λi($ \forall x\in {{\mathbb{R}}^{N}}$),且0<λi*=$ \mathop {\sup }\limits_{x \in {\mathbb{R}^N}} {\mkern 1mu} {\lambda _i}\left( x \right)$.则存在β*>0,使得当β∈(β*,+∞)时,系统(1)有正基态解(uv).

    注1  条件(H1)为引理4的证明提供了重要保证,当β>0充分大时,系统(1)没有半平凡解.

    为了方便,记C为不同的正常数,Lq($ {{\mathbb{R}}^{N}}$)和L($ {{\mathbb{R}}^{N}}$)中的范数分别记为‖·‖q和‖·‖on(1)表示当n→+∞时的无穷小量,Br(x)表示一个球心在x$ {{\mathbb{R}}^{N}}$且半径为r>0的开球.

    1 预备知识

    首先,回顾一下有关分数阶Laplace算子的基本概念.

    对于s∈(0,1),ζ=(ζ1ζ2,…,ζN),分数阶Laplace算子的可测函数u$ {{\mathbb{R}}^{N}}\to \mathbb{R}$定义为

    $ {\left( { - \Delta } \right)^s}u\left( x \right) = - \frac{1}{2}C\left( {N,s} \right)\int_{{\mathbb{R}^N}} {\frac{{u\left( {x + y} \right) + u\left( {x - y} \right) - 2u\left( x \right)}}{{{{\left| y \right|}^{1 + 2s}}}}{\text{d}}y} $

    其中

    $ C\left( {N,s} \right) = {\left( {\int_{{\mathbb{R}^N}} {\frac{{1 - \cos \left( {{\zeta _1}} \right)}}{{{{\left| \zeta \right|}^{N + 2s}}}}{\text{d}}\zeta } } \right)^{ - 1}} $

    Hs($ {{\mathbb{R}}^{N}}$)={uL2($ {{\mathbb{R}}^{N}}$):[u]s<+∞}为关于下面范数的分数阶Sobolev空间:

    $ \begin{array}{*{20}{c}} {{{\left\| u \right\|}_s} = {{\left( {\left[ u \right]_s^2 + \int_{{\mathbb{R}^N}} {{u^2}{\text{d}}x} } \right)}^{\frac{1}{2}}}}&{{{\left[ u \right]}_s} = {{\left( {\int_{{\mathbb{R}^N}} {\int_{{\mathbb{R}^N}} {\frac{{{{\left| {u\left( x \right) - v\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}{\text{d}}x{\text{d}}y} } } \right)}^{\frac{1}{2}}}} \end{array} $

    其中[u]s为关于函数u的Gagliardo半范数.同时,对$ \forall u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)$,有

    $ \left\| {{{\left( { - \Delta } \right)}^{\frac{s}{2}}}u} \right\|_2^2 = \frac{{C\left( {N,s} \right)}}{2}\int_{{\mathbb{R}^N}} {\int_{{\mathbb{R}^N}} {\frac{{{{\left| {u\left( x \right) - v\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}{\text{d}}x{\text{d}}y} } $

    为简单起见,本文将忽略归一化常数,用Ei表示的分数阶Sobolev空间的内积定义如下:

    $ {\left( {u,v} \right)_{{E_i}}} = \int_{{\mathbb{R}^N}} {{{\left( { - \Delta } \right)}^{\frac{s}{2}}}u{{\left( { - \Delta } \right)}^{\frac{s}{2}}}v{\text{d}}x} + \int_{{\mathbb{R}^N}} {{V_i}\left( x \right)uv{\text{d}}x} $

    相应的范数为$ {{\left\| u \right\|}_{{{E}_{i}}}}^{2}={{\left( u, u \right)}_{{{E}_{i}}}}$,且Ei连续地嵌入到Lp($ {{\mathbb{R}}^{N}}$)中,p ∈[2,2s*].记E=E1×E2,对∀(uv)∈E,有

    $ \left\| {\left( {u,v} \right)} \right\|_E^2 = \left\| {\left( u \right)} \right\|_{{E_1}}^2 + \left\| {\left( v \right)} \right\|_{{E_2}}^2 $

    此外,记E*E的对偶空间.

    为了考虑系统(1)的正基态解,在E上定义如下能量泛函:

    $ \mathit{\Phi }\left( {u,v} \right) = \frac{1}{2}\left\| {\left( {u,v} \right)} \right\|_E^2 - \frac{1}{p}\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| u \right|}^p} + {\lambda _2}\left( x \right){{\left| v \right|}^p}} \right){\text{d}}x} - \frac{{2\beta }}{p}\int_{{\mathbb{R}^N}} {{{\left| u \right|}^{\frac{p}{2}}}{{\left| v \right|}^{\frac{p}{2}}}{\text{d}}x} $ (3)

    易知,系统(1)的解与泛函ΦC2(E$ \mathbb{R}$)的临界点一一对应.

    类似于文献[13-14],定义与Φ相对应的Nehari流形如下:

    $ \begin{align} & \mathcal{N}=\left\{ \left( u,v \right)\in E\backslash \left\{ \left( 0,0 \right) \right\}:{\mathit{\Phi } }'\left( u,v \right)\left( u,v \right)=0 \right\}= \\ & \left\{ \left( u,v \right)\in E\backslash \left\{ \left( 0,0 \right) \right\}:\left\| \left( u,v \right) \right\|_{E}^{2}=\int_{{{\mathbb{R}}^{N}}}{\left( {{\lambda }_{1}}\left( x \right){{\left| u \right|}^{p}}+{{\lambda }_{2}}\left( x \right){{\left| v \right|}^{p}} \right)\text{d}x}+2\beta \int_{{{\mathbb{R}}^{N}}}{{{\left| u \right|}^{\frac{p}{2}}}{{\left| v \right|}^{\frac{p}{2}}}\text{d}x} \right\} \\ \end{align} $ (4)

    并且在$ \mathcal{N}$上,定义

    $ c = \mathop {\inf }\limits_{\left( {u,v} \right) \in \mathcal{N}} \mathit{\Phi }\left( {u,v} \right) $ (5)
    2 主要结果的证明

    为了证明定理1,本节给出一些主要的引理.

    引理1 假设条件(H1)-(H3)成立,则对∀z=(uv)∈E\{(0,0)},存在唯一的tz>0,使得tzz$ \mathcal{N}$.

     对每个固定的z=(uv)∈E\{(0,0)},定义φ(t)=Φ(tz)(t∈[0,+∞)),则由φ′(t)=0知tz$ \mathcal{N}$,那么

    $ \left\| {\left( {u,v} \right)} \right\|_E^2 = {t^{p - 2}}\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| u \right|}^p} + {\lambda _2}\left( x \right){{\left| v \right|}^p}} \right){\text{d}}x} + 2\beta {t^{p - 2}}\int_{{\mathbb{R}^N}} {{{\left| u \right|}^{\frac{p}{2}}}{{\left| v \right|}^{\frac{p}{2}}}{\text{d}}x} $ (6)

    其中t ∈(0,+∞).由条件(H1)-(H3)知φ(0)=0.当t>0充分小时,φ(t)>0;当t>0充分大时,φ(t)<0.且(6)式右端是一个关于t的严格增函数.则易知存在唯一的tz>0,使得φ(tz)=$ \mathop {\max }\limits_{t \geqslant 0} {\mkern 1mu} $φ(t)且φ′(tz)=0,也即是Φ(tzz)=$\mathop {\max }\limits_{t \geqslant 0} {\mkern 1mu} $Φ(tz)且tzz$ \mathcal{N}$.

    注2  在(H1)-(H3)成立的条件下,对∀(uv)∈$\mathcal{N}$,不难得到下列关系式成立:

    $ \begin{array}{*{20}{c}} {\mathit{\Phi }\left( {tu,tv} \right) \le \mathit{\Phi }\left( {u,v} \right)}&{\forall t \in \left[ {0, + \infty } \right)} \end{array} $
    $ c = \mathop {\inf }\limits_{\left( {u,v} \right) \in {\cal N}} \mathit{\Phi }\left( {u,v} \right) > 0 $

    引理2(山路几何结构)  假设条件(H1)-(H3)成立,容易验证泛函Φ满足下列条件:

    (ⅰ)存在常数ρ0α>0,如果‖(uv)‖E=ρ0,则Φ(uv)≥α

    (ⅱ)存在(u0v0)∈E,使得‖(u0v0)‖Eρ0Φ(u0v0)<0.

    Φ满足引理2的条件下,由文献[15]知,存在一个(Ce)c序列{(unvn)}⊂E,使得:

    $ \mathit{\Phi }\left( {{u_n},{v_n}} \right) \to c' = \mathop {\inf }\limits_{\gamma \in \mathit{\Gamma }} \mathop {\max }\limits_{t \in \left[ {0,1} \right]} \mathit{\Phi }\left( {\gamma \left( t \right)} \right) $
    $ \left( {1 + {{\left\| {\left( {{u_n},{v_n}} \right)} \right\|}_E}} \right){\left\| {\mathit{\Phi '}\left( {{u_n},{v_n}} \right)} \right\|_{{E^ * }}} \to 0 $

    其中

    $ \mathit{\Gamma } = \left\{ {\gamma \in C\left( {\left[ {0,1} \right],E} \right):\gamma \left( 0 \right) = \left( {0,0} \right),\mathit{\Phi }\left( {\gamma \left( 1 \right)} \right) < 0} \right\} $

    类似于文献[14]中引理2.4的证明,有

    $ c' = \mathop {\inf }\limits_{\left( {u,v} \right) \in E\backslash \left\{ {\left( {0,0} \right)} \right\}} \mathop {\max }\limits_{t \ge 0} \mathit{\Phi }\left( {tu,tv} \right) = c $

    引理3  假设条件(H1)-(H3)成立,如果对任意的(Ce)c序列{(unvn)}⊂E,满足:

    $ \begin{array}{*{20}{c}} {\mathit{\Phi }\left( {{u_n},{v_n}} \right) \to c}&{\left( {1 + {{\left\| {\left( {{u_n},{v_n}} \right)} \right\|}_E}} \right){{\left\| {\mathit{\Phi '}\left( {{u_n},{v_n}} \right)} \right\|}_{{E^ * }}} \to 0} \end{array} $ (7)

    则序列{(unvn)}在E中有界.

      设∀{(unvn)}⊂E且满足(7)式,那么

    $ \begin{array}{*{20}{c}} {c + {o_n}\left( 1 \right) = \mathit{\Phi }\left( {{u_n},{v_n}} \right) - \frac{1}{p}\mathit{\Phi '}\left( {{u_n},{v_n}} \right)\left( {{u_n},{v_n}} \right) = }\\ {\left( {\frac{1}{2} - \frac{1}{\rho }} \right)\left\| {\left( {{u_n},{v_n}} \right)} \right\|_E^2} \end{array} $

    即知{(unvn)}是有界的.

    引理4  假设条件(H1)-(H3)成立,如果(uv)是系统(1)的正基态解,且存在正常数β*>0,使得ββ*时有uv≢0.

      运用文献[9-10]的证明思想,不难证得下列单个方程可以分别获得正基态解u0v0

    $ \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\left( { - \Delta } \right)^s}u + {V_1}\left( x \right)u = {\lambda _1}\left( x \right){u^{p - 1}} \hfill \\ u > 0 \hfill \\ \end{array} &\begin{array}{l} x \in {\mathbb{R}^N} \hfill \\ x \in {\mathbb{R}^N} \hfill \\ \end{array} \end{array}} \right. $
    $ \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\left( { - \Delta } \right)^s}v + {V_2}\left( x \right)v = {\lambda _2}\left( x \right){v^{p - 1}} \hfill \\ v > 0 \hfill \\ \end{array} &\begin{array}{l} x \in {\mathbb{R}^N} \hfill \\ x \in {\mathbb{R}^N} \hfill \\ \end{array} \end{array}} \right. $

    为证引理4,只需证

    $ c < \min \left\{ {\mathit{\Phi }\left( {{u_0},0} \right),\mathit{\Phi }\left( {0,{v_0}} \right)} \right\} $ (8)

    由引理1知,存在常数t0>0,使得(t0u0t0v0)∈$ \mathcal{N}$,有

    $ \begin{gathered} t_0^2\left\| {\left( {{u_0},{v_0}} \right)} \right\|_E^2 = t_0^p\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{u_0}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{v_0}} \right|}^p}} \right){\text{d}}x} + 2\beta t_0^p\int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} \geqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\beta t_0^p\int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} \hfill \\ \end{gathered} $

    $ {t_0} \leqslant {\left( {\frac{{\left\| {\left( {{u_0},{v_0}} \right)} \right\|_E^2}}{{2\beta \int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} }}} \right)^{\frac{1}{{p - 2}}}} = {\beta ^{ - \frac{1}{{p - 2}}}}M $ (9)

    再由(4),(5),(9)式和φ(t)的定义可得

    $ \begin{array}{l} c \leqslant \mathit{\Phi }\left( {{t_0}{u_0},{t_0}{v_0}} \right) = \hfill \\ \;\;\;\;\;\frac{{t_0^2}}{2}\left\| {\left( {{u_0},{v_0}} \right)} \right\|_E^2 - \frac{{t_0^p}}{p}\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{u_0}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{v_0}} \right|}^p}} \right){\text{d}}x} - \frac{{2\beta }}{p}t_0^p\int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} = \hfill \\ \;\;\;\;\;\left( {\frac{1}{2} - \frac{1}{p}} \right)t_0^p\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{u_0}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{v_0}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right)\beta t_0^p\int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} \leqslant \hfill \\ \;\;\;\;\;\left( {\frac{1}{2} - \frac{1}{p}} \right){\beta ^{ - \frac{p}{{p - 2}}}}{M^p}\int_{{\mathbb{R}^N}} {\left( {\lambda _1^ * {{\left| {{u_0}} \right|}^p} + \lambda _2^ * {{\left| {{v_0}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right){\beta ^{ - \frac{2}{{p - 2}}}}{M^p}\int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} \leqslant \hfill \\ \;\;\;\;\;{\beta ^{ - \frac{p}{{p - 2}}}}{M^p}C\int_{{\mathbb{R}^N}} {\left( {{{\left| {{u_0}} \right|}^p} + {{\left| {{v_0}} \right|}^p}} \right){\text{d}}x} + {\beta ^{ - \frac{p}{{p - 2}}}}{M^p}C\int_{{\mathbb{R}^N}} {{{\left| {{u_0}} \right|}^{\frac{p}{2}}}{{\left| {{v_0}} \right|}^{\frac{p}{2}}}{\text{d}}x} = \hfill \\ \;\;\;\;\;g\left( \beta \right) \hfill \\ \end{array} $ (10)

    要证(8)式成立,即需证

    $ g\left( \beta \right) < \min \left\{ {\mathit{\Phi }\left( {{u_0},0} \right),\mathit{\Phi }\left( {0,{v_0}} \right)} \right\} $

    注意到在(H1)的条件下,在(10)式中,当β→+∞时g(β)→0.则存在常数β*>0,使得当ββ*时(8)式成立.

    定理1的证明  由引理3可知,存在有界的(Ce)c序列{(unvn)}E满足(7)式.现假设

    $ \sigma = \mathop {\lim \sup }\limits_{n \to + \infty } \mathop {\sup }\limits_{y \in {\mathbb{R}^N}} \int_{{B_1}\left( y \right)} {\left( {{{\left| {{u_n}} \right|}^2} + {{\left| {{v_n}} \right|}^2}} \right){\text{d}}x} = 0 $

    则由分数阶Sobolev空间的集中紧性原理(参见文献[9]中引理Ⅱ.4),对∀p∈(2,2s*),在Lp($ {{\mathbb{R}}^{N}}$Lp($ {{\mathbb{R}}^{N}}$)上有(unvn)→(0,0),那么

    $ \begin{array}{l} 0 < c = \mathit{\Phi }\left( {{u_n},{v_n}} \right) = \frac{1}{2}\mathit{\Phi '}\left( {{u_n},{v_n}} \right)\left( {{u_n},{v_n}} \right) + o\left( 1 \right) = \hfill \\ \;\;\;\;\;\left( {\frac{1}{2} - \frac{1}{p}} \right)\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{u_n}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{v_n}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right)\beta \int_{{\mathbb{R}^N}} {{{\left| {{u_n}} \right|}^{\frac{p}{2}}}{{\left| {{v_n}} \right|}^{\frac{p}{2}}}{\text{d}}x} + {o_n}\left( 1 \right) \geqslant \hfill \\ \;\;\;\;\;\left( {\frac{1}{2} - \frac{1}{p}} \right)C\int_{{\mathbb{R}^N}} {\left( {{{\left| {{u_n}} \right|}^p} + {{\left| {{v_n}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right)\beta \int_{{\mathbb{R}^N}} {{{\left| {{u_n}} \right|}^{\frac{p}{2}}}{{\left| {{v_n}} \right|}^{\frac{p}{2}}}{\text{d}}x} + {o_n}\left( 1 \right) = \hfill \\ \;\;\;\;\;{o_n}\left( 1 \right) \hfill \\ \end{array} $

    矛盾,从而σ>0.

    必要时取子序列(仍记为{(unvn)}),那么存在常数R>0和{yn}⊂$ {{\mathbb{Z}}^{N}}$,使得

    $ \int_{{B_R}\left( 0 \right)} {\left( {{{\left| {{{\tilde u}_n}} \right|}^2} + {{\left| {{{\tilde v}_n}} \right|}^2}} \right){\text{d}}x} = \int_{{B_R}\left( {{y_n}} \right)} {\left( {{{\left| {{u_n}} \right|}^2} + {{\left| {{v_n}} \right|}^2}} \right){\text{d}}x} > \frac{\sigma }{2} $ (11)

    其中

    $ \left( {{{\tilde u}_n}\left( x \right),{{\tilde v}_n}\left( x \right)} \right) = \left( {{u_n}\left( {x + {y_n}} \right),{v_n}\left( {x + {y_n}} \right)} \right) $

    又由Φ$ \mathcal{N}$的平移不变性知:

    $ \begin{array}{*{20}{c}} {\mathit{\Phi }\left( {{{\tilde u}_n},{{\tilde v}_n}} \right) \to c}&{\left( {1 + {{\left\| {\left( {{{\tilde u}_n},{{\tilde v}_n}} \right)} \right\|}_E}} \right)} \end{array}{\left\| {\mathit{\Phi '}\left( {{{\tilde u}_n},{{\tilde v}_n}} \right)} \right\|_{{E^ * }}} \to 0 $

    必要时再取一个子序列,存在($ \tilde{u}, \tilde{v}$)∈E使得

    $ \left\{ \begin{array}{l} \left( {{{\tilde u}_n},{{\tilde v}_n}} \right) \rightharpoonup \left( {\tilde u,\tilde v} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in E \hfill \\ \left( {{{\tilde u}_n},{{\tilde v}_n}} \right) \to \left( {\tilde u,\tilde v} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in L\left( {{B_R}\left( 0 \right)} \right) \times L\left( {{B_R}\left( 0 \right)} \right) \hfill \\ \left( {{{\tilde u}_n}\left( x \right),{{\tilde v}_n}\left( x \right)} \right) \to \left( {\tilde u\left( x \right),\tilde v\left( x \right)} \right)\;\;\;\;\;\;\;\;{\text{a}}.\;{\text{e}}.\;\;x \in {\mathbb{R}^N} \hfill \\ \end{array}\right. $

    则由(11)式可得($ \tilde{u}, \tilde{v}$)≠(0,0),再由一般的讨论方法知,Φ′($ \tilde{u}, \tilde{v}$)=0,($ \tilde{u}, \tilde{v}$)∈$ \mathcal{N}$c≤Φ($ \mathcal{N}$).

    此外,由Fatou引理和条件(H2)有

    $ \begin{array}{l} c = \mathop {\lim \inf }\limits_{n \to + \infty } \left[ {\mathit{\Phi }\left( {{u_n},{v_n}} \right) - \frac{1}{2}\mathit{\Phi '}\left( {{u_n},{v_n}} \right)\left( {{u_n},{v_n}} \right)} \right] = \hfill \\ \;\;\;\;\;\mathop {\lim \inf }\limits_{n \to + \infty } \left[ {\left( {\frac{1}{2} - \frac{1}{p}} \right)\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{u_n}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{v_n}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right)\beta \int_{{\mathbb{R}^N}} {{{\left| {{u_n}} \right|}^{\frac{p}{2}}}{{\left| {{v_n}} \right|}^{\frac{p}{2}}}{\text{d}}x} } \right] = \hfill \\ \;\;\;\;\;\mathop {\lim \inf }\limits_{n \to + \infty } \left[ {\left( {\frac{1}{2} - \frac{1}{p}} \right)\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{{\tilde u}_n}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{{\tilde v}_n}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right)\beta \int_{{\mathbb{R}^N}} {{{\left| {{{\tilde u}_n}} \right|}^{\frac{p}{2}}}{{\left| {{{\tilde v}_n}} \right|}^{\frac{p}{2}}}{\text{d}}x} } \right] \geqslant \hfill \\ \;\;\;\;\;\left( {\frac{1}{2} - \frac{1}{p}} \right)\int_{{\mathbb{R}^N}} {\left( {{\lambda _1}\left( x \right){{\left| {{{\tilde u}}} \right|}^p} + {\lambda _2}\left( x \right){{\left| {{{\tilde v}}} \right|}^p}} \right){\text{d}}x} + \left( {1 - \frac{2}{p}} \right)\beta \int_{{\mathbb{R}^N}} {{{\left| {{{\tilde u}}} \right|}^{\frac{p}{2}}}{{\left| {{{\tilde v}}} \right|}^{\frac{p}{2}}}{\text{d}}x} \hfill \\ \;\;\;\;\; = \mathit{\Phi }\left( {{{\tilde u}},{{\tilde v}}} \right) - \frac{1}{2}\mathit{\Phi '}\left( {\tilde u,\tilde v} \right)\left( {\tilde u,\tilde v} \right) = \hfill \\ \;\;\;\;\;\mathit{\Phi }\left( {\tilde u,\tilde v} \right) \hfill \\ \end{array} $

    那么c=Φ($ \tilde{u}, \tilde{v}$),又由引理4知,当ββ*时,$ \tilde{u}, \tilde{v}$≢0.则($ \tilde{u}, \tilde{v}$)是系统(1)的非平凡基态解.

    下面证明系统(1)存在正基态解.由引理1知,存在$ {\tilde{t}}$>0,使得:

    $ \begin{array}{*{20}{c}} {\left( {\tilde t\left| {\tilde u} \right|,\tilde t\left| {\tilde v} \right|} \right) \in {\cal N}}&{\mathit{\Phi }\left( {\tilde t\left| {\tilde u} \right|,\tilde t\left| {\tilde v} \right|} \right) \ge c} \end{array} $

    又由条件(H1)-(H3)、注2及泛函Φ的形式有

    $ \mathit{\Phi }\left( {\tilde t\left| {\tilde u} \right|,\tilde t\left| {\tilde v} \right|} \right) \le \mathit{\Phi }\left( {\tilde t\tilde u,\tilde t\tilde v} \right) \le \mathit{\Phi }\left( {\tilde u,\tilde v} \right) $

    则($ \tilde{t}\left| {\tilde{u}} \right|, \tilde{t}\left| {\tilde{v}} \right|$)=(uv)是系统(1)的非平凡基态解.最后由强极大值原理(参见文献[16])可得,对∀x$ {{\mathbb{R}}^{N}}$都有uv>0.

    参考文献
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    Positive Ground State Solutions for a Class of Fractional Coupled System with Periodic Potentials
    HE Shu-wen, SHANG Yan-ying     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: In this article, we investigate the existence of nontrivial solutions for a class of fractional coupled system with different periodic potentials by using the Ekeland's variational principle. The point is to prove that the nontrivial solutions of the system can be a positive ground state solution, which extends the general Schrödinger coupled system to the case of the fractional coupled system with multiple different periodic functions.
    Key words: fractional coupled system    Ekeland's variational principle    positive ground state solution    
    X