西南大学学报 (自然科学版)  2018, Vol. 40 Issue (10): 89-94.  DOI: 10.13718/j.cnki.xdzk.2018.10.015
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  • 带有凹凸非线性项的Kirchhoff型方程解的多重性    [PDF全文]
    王雅琪, 欧增奇     
    西南大学 数学与统计学院, 重庆 400715
    摘要:利用集中紧性原理和对偶喷泉定理,研究了一类带有凹凸非线性项的Kirchhoff方程 $ \left\{ \begin{array}{l} - \left( {a + b\int_\mathit{\Omega} {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u = {\left| u \right|^4}u + \mu {\left| u \right|^{q - 2}}u\;\;\;\;\;\;\;x \in \mathit{\Omega} \\ u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x \in \partial \mathit{\Omega} \end{array} \right. $ 获得了该方程有无穷多个解.其中Ω$ \mathbb{R} $3中边界光滑的有界开集,且ab > 0,1 < q < 2,μ > 0.
    关键词Kirchhoff方程    凹凸非线性项    集中紧性原理    对偶喷泉定理    

    考虑如下Kirchhoff方程:

    $ \left\{ {\begin{array}{*{20}{l}} { - \left( {a + b\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\nabla u = {{\left| u \right|}^4}u + \mu {{\left| u \right|}^{q - 2}}u}&{x \in \mathit{\Omega }}\\ {u = 0}&{x \in \partial \mathit{\Omega }} \end{array}} \right. $ (1)

    其中Ω$ \mathbb{R} $3中边界光滑的有界开集,且ab>0,1<q<2,μ>0.我们记Sobolev空间H01(Ω)中的范数为

    $ \left\| u \right\| = {\left( {\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)^{\frac{1}{2}}} $

    Ls(Ω)中的范数为

    $ {\left| u \right|_s} = {\left( {\int_\mathit{\Omega } {{{\left| u \right|}^s}{\rm{d}}x} } \right)^{\frac{1}{s}}} $

    当1≤s≤6时,嵌入H01(Ω)↺Ls(Ω)是连续的;当1≤s<6时,嵌入是紧的.此外,最佳Sobolev常数为

    $ S = \mathop {{\rm{inf}}}\limits_{u \in H_0^1\left( \mathit{\Omega } \right)\backslash \left\{ 0 \right\}} \frac{{\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} }}{{{{\left( {\int_\mathit{\Omega } {{{\left| u \right|}^6}{\rm{d}}x} } \right)}^{\frac{1}{3}}}}} $ (2)

    由于有bΩ|∇u|2dx这一项,方程(1)被称为非局部问题.众所周知,Kirchhoff型问题起源于文献[1],作为经典的D′Alembert波动方程在弹性弦的自由振动的推广.文献[2]给出一个泛函分析结构,Kirchhoff型问题逐渐引起人们的关注.据我们查阅的文献显示,文献[3]最先将变分法运用到Kirchhoff型问题中.此后,出现了诸多关于Kirchhoff型问题的优秀结论[1-2, 4-9].

    N=3时,文献[4-5, 10]研究了Kirchhoff型方程正解的存在性和多重性.文献[10]研究了0<q<1时的情形,利用Nehari和Ekeland变分原则的方法,得到了“存在一个仅依赖于aT4(a)>0,当a>0,0<λT4(a)时,方程至少有一个正解”的结论.当b充分小时,文献[4]利用极小作用原理和山路引理的方法,获得了方程(1)的两个正解.文献[5]研究了a=1,q=2时的情形,证得方程(1)具有正的基态解.受到文献[4-6, 10-11]的启发,本文将研究$ \mathbb{R} $3空间中方程(1)多解的存在情况,并得到下面的定理:

    定理 1  假设Ω$ \mathbb{R} $3有界,并且ab>0,1<q<2,则存在μ*>0,使得对∀0<μμ*,方程(1)有一列解{un},并且φμ(un)<0,φμ(un)→0(n→∞).

    我们定义φμ(u)为方程(1)对应的能量泛函,即

    $ \begin{array}{*{20}{c}} {{\varphi _\mu }\left( u \right) = \frac{a}{2}{{\left\| u \right\|}^2} + \frac{b}{4}{{\left\| u \right\|}^4} - \frac{1}{6}\int_\mathit{\Omega } {{{\left| u \right|}^6}{\rm{d}}x} - \frac{\mu }{q}\int_\mathit{\Omega } {{{\left| u \right|}^q}{\rm{d}}x} }&{\forall u \in H_0^1\left( \mathit{\Omega } \right)} \end{array} $ (3)

    如果uH01(Ω),且对∀vH01(Ω),都有

    $ a\int_\mathit{\Omega } {\nabla u\nabla v{\rm{d}}x} + b{\left\| u \right\|^2}\int_\mathit{\Omega } {\nabla u\nabla v{\rm{d}}x} - \int_\mathit{\Omega } {{{\left| u \right|}^4}uv{\rm{d}}x} - \mu \int_\mathit{\Omega } {{{\left| u \right|}^{q - 2}}uv{\rm{d}}x} = 0 $

    u为方程(1)的弱解.

    X是自反的可分Banach空间,则存在eiXej*=X*,使得:

    $ \begin{array}{*{20}{c}} {X = \overline {{\rm{span}}\left\{ {{e_i}:i = 1,2, \cdots } \right\}} }&{{X^*} = \overline {{\rm{span}}\left\{ {{e_j}:j = 1,2, \cdots } \right\}} } \end{array} $

    $ \left\langle {e_j^*,{e_i}} \right\rangle = \left\{ {\begin{array}{*{20}{l}} 1&{i = j}\\ 0&{i \ne j} \end{array}} \right. $

    Xj=span{ej},于是$ X = \overline {{ \otimes _{j \ge 1}}{X_j}} $.记Yk=⊕j=1kXj$ {Z_k} = \overline {{ \oplus _{j \ge k}}{X_j}} $.

    引理 1  假设abμ>0,1<q<2,以及$ c < \mathit{\Lambda} - D{ \mu^{\frac{2}{{2 - q}}}} $,则泛函φμ满足局部(PS)c*条件,其中:

    $ \mathit{\Lambda } = \frac{1}{4}ab{S^3} + \frac{1}{{24}}{b^3}{S^6} + \frac{1}{{24}}{\left( {{b^2}{S^4} + 4aS} \right)^{\frac{3}{2}}} $
    $ D = \frac{{2 - q}}{2}{\left( {\frac{1}{q} - \frac{1}{4}} \right)^{\frac{2}{{2 - q}}}}{\left| \mathit{\Omega } \right|^{\frac{{6 - q}}{{6 - 3q}}}}{\left( {\frac{{2q}}{{aS}}} \right)^{\frac{q}{{2 - q}}}} $

      取H01(Ω)中的标准正交基(ej),并且定义Xj=$ \mathbb{R} $ej.假设{unj}是泛函φμ的(PS)c*序列,即

    $ \begin{array}{*{20}{c}} {{u_{{n_j}}} \in {Y_{{n_j}}},{\varphi _\mu }\left( {{u_{{n_j}}}} \right) \to c,{\varphi _\mu }{{|'}_{{Y_{{n_j}}}}}\left( {{u_{{n_j}}}} \right) \to 0}&{{n_j} \to \infty } \end{array} $ (4)

    现证明{unj}在H01(Ω)中有收敛子列.首先,由(3),(4)式、Hölder不等式以及Sobolev不等式,有

    $ \begin{array}{l} 1 + c + o\left( {\left\| {{u_{{n_j}}}} \right\|} \right) \ge {\varphi _\mu }\left( {{u_{{n_j}}}} \right) - \frac{1}{6}\left\langle {{\varphi _\mu }\left( {{u_{{n_j}}}} \right),{u_{{n_j}}}} \right\rangle \ge \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{a}{3}{\left\| {{u_{{n_j}}}} \right\|^2} + \frac{b}{{12}}{\left\| {{u_{{n_j}}}} \right\|^4} - \frac{\mu }{{{S^{\frac{q}{2}}}}}\left( {\frac{1}{q} - \frac{1}{6}} \right){\left| \mathit{\Omega } \right|^{\frac{{6 - q}}{6}}}{\left\| {{u_{{n_j}}}} \right\|^q} \end{array} $ (5)

    由于1<q<2,根据(5)式,可知{unj}在H01(Ω)中有界.因此,存在{unj}的子列(不妨仍记为{unj})以及uH01(Ω),使得

    $ \left\{ {\begin{array}{*{20}{l}} {{u_{{n_j}}} \rightharpoonup u}&{x \in H_0^1\left( \mathit{\Omega } \right)}\\ {{u_{{n_j}}} \to u}&{x \in {L^p}\left( \mathit{\Omega } \right),1 \le p < 6}\\ {{u_{{n_j}}}\left( x \right) \to u\left( x \right)}&{{\rm{a}}{\rm{.e}}{\rm{.}}\;\;\;x \in \mathit{\Omega }} \end{array}} \right. $

    根据第二集中性引理[12],我们可以找到一个至多可数的指标集Γ、在$ \mathbb{R} $3中的一个序列{xk}kΓ,以及{ηk}kΓ,{νk}kΓ$ \mathbb{R} $+,使得:

    $ {\left| {\nabla {u_{{n_j}}}} \right|^2} \rightharpoonup {\rm{d}}\eta \ge {\left| {\nabla u} \right|^2} + \sum\limits_{k \in \mathit{\Gamma }} {{\eta _k}{\delta _{{x_k}}}} $ (6)
    $ {\left| {{u_{{n_j}}}} \right|^6} \rightharpoonup {\rm{d}}\nu {\rm{ = }}{\left| u \right|^6} + \sum\limits_{k \in \mathit{\Gamma }} {{\nu _k}{\delta _{{x_k}}}} $ (7)
    $ {\eta _k} \ge S\nu _k^{\frac{1}{3}} $ (8)

    其中δxk是在xk上的Diracdelta函数.接下来,我们证明Γ=.假设Γ,不妨设kΓ,对∀ε>0,设ψεkC0($ \mathbb{R} $3,[0, 1])满足条件0≤ψεk≤1,|∇ψεk|≤C,且:

    $ \left\{ {\begin{array}{*{20}{l}} {\psi _\varepsilon ^k\left( x \right) \equiv 1}&{x \in {B_\varepsilon }\left( {{x_k}} \right)}\\ {\psi _\varepsilon ^k\left( x \right) \equiv 0}&{x \in \mathit{\Omega \backslash }{\mathit{B}_{2\varepsilon }}\left( {{x_k}} \right)} \end{array}} \right. $

    由于{ψεkunj}在H01(Ω)上有界,我们有〈φμ′(unj),ψεkunj〉→0,即

    $ \begin{array}{l} \left( {a + b{{\left\| {{u_{{n_j}}}} \right\|}^2}} \right)\left( {\int_\mathit{\Omega } {{u_{{n_j}}}\nabla {u_{{n_j}}}\nabla \psi _\varepsilon ^k{\rm{d}}x} + {{\int_\mathit{\Omega } {\left| {\nabla {u_{{n_j}}}} \right|} }^2}\psi _\varepsilon ^k{\rm{d}}x} \right) = \\ {\int_\mathit{\Omega } {\left| {{u_{{n_j}}}} \right|} ^6}\psi _\varepsilon ^k{\rm{d}}x + \mu {\int_\mathit{\Omega } {\left| {{u_{{n_j}}}} \right|} ^q}\psi _\varepsilon ^k{\rm{d}}x + o\left( 1 \right) \end{array} $ (9)

    由于{unj}在H01(Ω)上有界,并且由Hölder不等式,则存在常数C1C2C3>0,有

    $ \begin{array}{l} \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \left( {a + b{{\left\| {{u_{{n_j}}}} \right\|}^2}} \right)\left| {\int_\mathit{\Omega } {{u_{{n_j}}}\nabla {u_{{n_j}}}\nabla \psi _\varepsilon ^k{\rm{d}}x} } \right| \le \\ \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } {C_1}{\left( {\int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {{{\left| {\nabla {u_{{n_j}}}} \right|}^2}} {\rm{d}}x} \right)^{\frac{1}{2}}}{\left( {{{\int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {\left| {\nabla \psi _\varepsilon ^k} \right|} }^2}{{\left| {{u_{{n_j}}}} \right|}^2}{\rm{d}}x} \right)^{\frac{1}{2}}} \le \\ \mathop {\lim }\limits_{\varepsilon \to 0} {C_2}{\left( {{{\int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {\left| {\nabla \psi _\varepsilon ^k} \right|} }^2}{{\left| u \right|}^2}{\rm{d}}x} \right)^{\frac{1}{2}}} \le \\ \mathop {\lim }\limits_{\varepsilon \to 0} {C_2}{\left( {{{\int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {\left| {\nabla \psi _\varepsilon ^k} \right|} }^3}{\rm{d}}x} \right)^{\frac{1}{3}}}{\left( {\int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {{{\left| u \right|}^6}} {\rm{d}}x} \right)^{\frac{1}{6}}} \le \\ \mathop {\lim }\limits_{\varepsilon \to 0} {C_3}{\left( {\int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {{{\left| u \right|}^6}} {\rm{d}}x} \right)^{\frac{1}{6}}} = 0 \end{array} $

    从而

    $ \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \left( {a + b{{\left\| {{u_{{n_j}}}} \right\|}^2}} \right)\int_\mathit{\Omega } {{u_{{n_j}}}\nabla {u_{{n_j}}}\nabla \psi _\varepsilon ^k{\rm{d}}x} = 0 $ (10)

    由(6)式,我们可知

    $ \begin{array}{l} \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \left( {a + b{{\left\| {{u_{{n_j}}}} \right\|}^2}} \right)\int_\mathit{\Omega } {{{\left| {\nabla {u_{{n_j}}}} \right|}^2}\psi _\varepsilon ^k} {\rm{d}}x \ge \\ \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \left( {a + b\int_\mathit{\Omega } {{{\left| {\nabla {u_{{n_j}}}} \right|}^2}\psi _\varepsilon ^k} {\rm{d}}x} \right)\int_\mathit{\Omega } {{{\left| {\nabla {u_{{n_j}}}} \right|}^2}\psi _\varepsilon ^k} {\rm{d}}x = \\ \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \int_\mathit{\Omega } {{{\left| {\nabla {u_{{n_j}}}} \right|}^2}\psi _\varepsilon ^k{\rm{d}}x} + \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } b{\left( {\int_\mathit{\Omega } {{{\left| {\nabla {u_{{n_j}}}} \right|}^2}\psi _\varepsilon ^k} {\rm{d}}x} \right)^2} \ge \\ a{\eta _k} + b{\eta _k}^2 \end{array} $ (11)

    由(7)式得

    $ \begin{array}{l} \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^6}} \psi _\varepsilon ^k{\rm{d}}x = \mathop {\lim }\limits_{\varepsilon \to 0} \int_\mathit{\Omega } {{{\left| u \right|}^6}} \psi _\varepsilon ^k{\rm{d}}x + {\nu _k} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\lim }\limits_{\varepsilon \to 0} \int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {{{\left| u \right|}^6}} \psi _\varepsilon ^k{\rm{d}}x + {\nu _k} = {\nu _k} \end{array} $ (12)

    由(12)式得

    $ \mathop {\lim }\limits_{\varepsilon \to 0} \mathop {{\rm{lim}}\;{\rm{sup}}}\limits_{{n_j} \to \infty } \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^q}} \psi _\varepsilon ^k{\rm{d}}x = \mathop {\lim }\limits_{\varepsilon \to 0} \int_{{\mathit{B}_\varepsilon }\left( {{x_k}} \right)} {{{\left| u \right|}^q}} \psi _\varepsilon ^k{\rm{d}}x = 0 $ (13)

    由(10)-(13)式,可推得

    $ {\nu _k} \ge a{\eta _k} + b{\eta _k}^2 $ (14)

    和(8)式比较,可得:

    (ⅰ) ηk=0;

    (ⅱ) $ {{\eta }_{k}}\ge \frac{b{{S}^{3}}+S\sqrt{{{b}^{2}}{{S}^{4}}+4aS}}{2} $.

    我们证明(ⅱ)不成立.根据文献[13]的引理2.2、Hölder不等式、Sobolev不等式,以及(6),(7),(14)式,可得

    $ \begin{array}{l} c = \mathop {\lim }\limits_{{n_j} \to \infty } \left( {{\varphi _\mu }\left( {{u_{{n_j}}}} \right)\frac{1}{4}\left\langle {{{\varphi '}_\mu }\left( {{u_{{n_j}}}} \right),{u_{{n_j}}}} \right\rangle } \right) = \\ \;\;\;\;\;\mathop {\lim }\limits_{{n_j} \to \infty } \left( {\frac{a}{4}{{\left\| {{u_{{n_j}}}} \right\|}^2} + \frac{1}{{12}}\int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^6}} {\rm{d}}x - \left( {\frac{1}{q} - \frac{1}{4}} \right)\mu \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^q}} {\rm{d}}x} \right) \ge \\ \;\;\;\;\;\frac{a}{4}{\left\| u \right\|^2} + \frac{a}{4}{\eta _k} + \frac{1}{{12}}{\nu _k} + \frac{1}{{12}}\int_\mathit{\Omega } {{{\left| u \right|}^6}} {\rm{d}}x - \left( {\frac{1}{q} - \frac{1}{4}} \right)\mu \int_\mathit{\Omega } {{{\left| u \right|}^q}} {\rm{d}}x \ge \\ \;\;\;\;\;\frac{a}{4}{\eta _k} + \frac{1}{{12}}{\nu _k} + \frac{a}{4}{\left\| u \right\|^2} - \left( {\frac{1}{q} - \frac{1}{4}} \right)\mu \int_\mathit{\Omega } {{{\left| u \right|}^q}} {\rm{d}}x \ge \\ \;\;\;\;\;\frac{a}{3}{\eta _k} + \frac{b}{{12}}\eta _k^2 + \frac{{aS}}{4}\left| u \right|_6^2 - \mu \left( {\frac{1}{q} - \frac{1}{4}} \right){\left| \mathit{\Omega } \right|^{\frac{{6 - q}}{6}}}\left| u \right|_6^q \end{array} $

    若(ⅱ)成立,则

    $ \frac{a}{3}{\eta _k} + \frac{b}{{12}}\eta _k^2 \ge \frac{1}{4}ab{S^3} + \frac{1}{{24}}{b^3}{S^6} + \frac{1}{{24}}{\left( {{b^2}{S^4} + 4aS} \right)^{\frac{3}{2}}} $

    为了估计$ \frac{aS}{4}\left| u \right|_{6}^{2}-\mu \left( \frac{1}{q}-\frac{1}{4} \right)|\mathit{\Omega }{{|}^{\frac{6-q}{6}}}\left| u \right|_{6}^{q} $,我们考虑

    $ \begin{array}{*{20}{c}} {f\left( t \right) = \frac{{aS}}{4}{t^2} - \mu \left( {\frac{1}{q} - \frac{1}{4}} \right){{\left| \mathit{\Omega } \right|}^{\frac{{6 - q}}{6}}}{t^q}}&{t \ge 0} \end{array} $ (15)

    得到$ \mathop {{\rm{min}}}\limits_{t \ge 0} f\left( t \right) = f({t_1}) = - D{\mu ^{\frac{2}{{2 - q}}}} $,其中:

    $ \begin{array}{*{20}{c}} {{t_1} = {{\left[ {\frac{{2q\mu }}{{aS}}\left( {\frac{1}{q} - \frac{1}{4}} \right){{\left| \mathit{\Omega } \right|}^{\frac{{6 - q}}{6}}}} \right]}^{\frac{1}{{2 - q}}}}}&{D = \frac{{1 - q}}{2}{{\left( {\frac{1}{q} - \frac{1}{4}} \right)}^{\frac{2}{{2 - q}}}}} \end{array}{\left| \mathit{\Omega } \right|^{\frac{{6 - q}}{{6 - 3q}}}}{\left( {\frac{{2q}}{{aS}}} \right)^{\frac{q}{{2 - q}}}} $

    因此,由(13)-(15)式,可知

    $ c \ge \frac{a}{3}{\eta _k} + \frac{b}{{12}}\eta _k^2 + \frac{{aS}}{4}\left| u \right|_6^2 - \mu \left( {\frac{1}{q} - \frac{1}{4}} \right){\left| \mathit{\Omega } \right|^{\frac{{6 - q}}{6}}}\left| u \right|_6^q \ge \mathit{\Lambda } - \mathit{D}{\mu ^{\frac{q}{{2 - q}}}} $

    故矛盾,所以(ⅱ)不成立,则ηk=0,即Γ=.所以我们可得出结论

    $ \mathop {\lim }\limits_{{n_j} \to \infty } \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^6}} {\rm{d}}x = \int_\mathit{\Omega } {{{\left| u \right|}^6}} {\rm{d}}x $

    接下来证明在H01(Ω)中unju.不妨设$ \mathop {{\rm{lim}}}\limits_{{n_j} \to \infty } {\left\| {{u_{{n_j}}}} \right\|^2} = {d^2} $,则需证‖u2=d2.事实上,

    $ \begin{array}{l} 0 = \mathop {\lim }\limits_{{n_j} \to \infty } \left\langle {{{\varphi '}_\mu }\left( {{u_{{n_j}}}} \right),{u_{{n_j}}} - u} \right\rangle = \mathop {\lim }\limits_{{n_j} \to \infty } {{\varphi '}_\mu }\left( {{u_{{n_j}}}} \right) - \mathop {\lim }\limits_{{n_j} \to \infty } {{\varphi '}_\mu }\left( {{u_{{n_j}}}} \right)u = \\ \;\;\;\;\;\mathop {\lim }\limits_{{n_j} \to \infty } \left[ {\left( {a + b{{\left\| {{u_{{n_j}}}} \right\|}^2}} \right){{\left\| {{u_{{n_j}}}} \right\|}^2} - \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^6}} {\rm{d}}x - \mu \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^q}} {\rm{d}}x} \right] - \\ \;\;\;\;\;\mathop {\lim }\limits_{{n_j} \to \infty } \left[ {\left( {a + b{{\left\| {{u_{{n_j}}}} \right\|}^2}} \right)\int_\mathit{\Omega } {\nabla {u_{{n_j}}}\nabla u} {\rm{d}}x - \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^4}} {u_{{n_j}}}u{\rm{d}}x - \mu \int_\mathit{\Omega } {{{\left| {{u_{{n_j}}}} \right|}^{q - 2}}{u_{{n_j}}}u} {\rm{d}}x} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {a + b{d^2}} \right)\left( {{d^2} - \int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right) \end{array} $

    因此,unju(xH01(Ω)).所以,当$c < \mathit{\Lambda} - D{\mu ^{\frac{2}{{2 - q}}}} $时,泛函φμ满足局部(PS)c*条件.

    定理1的证明  我们将用文献[14]中的对偶喷泉定理证明定理1.下面将证明对∀kk0,存在ρkγk>0,使得:

    $ ({{\rm{B}}_1}) {a_k} = \mathop {{\rm{inf}}}\limits_{u \in {Z_k}, \left\| u \right\| = {\rho _k}} \varphi (u) \ge 0; $

    $ ({{\rm{B}}_2}) {b_k} = \mathop {{\rm{max}}}\limits_{u \in {Y_k}, \left\| u \right\| = {\gamma _k}} \varphi (u) < 0; $

    $ ({{\rm{B}}_3})\;{d_k} = \mathop {{\rm{inf}}}\limits_{u \in {Z_k}, {\rm{ }}\left\| u \right\| = {\rho _k}} \to 0, \;k \to \infty . $

    事实上,为了证明条件(B1),我们定义${\beta _k} = \mathop {{\rm{sup}}}\limits_{u \in {Z_k}, \left\| u \right\| = 1} {\left| u \right|_q} $.由文献[6]的引理3.8,有βk→0(k→∞).同时存在R>0,使得

    $ \left\| u \right\| \le R \Rightarrow \frac{1}{{6{S^3}}}{\left\| u \right\|^6} \le \frac{a}{4}{\left\| u \right\|^2} $

    那么当uZk,‖u‖≤R时,有

    $ \begin{array}{l} {\varphi _\mu }\left( u \right) = \frac{a}{2}{\left\| u \right\|^2} + \frac{b}{4}{\left\| u \right\|^4} - \frac{1}{6}\int_\mathit{\Omega } {{{\left| u \right|}^6}} {\rm{d}}x - \frac{\mu }{q}\int_\mathit{\Omega } {{{\left| u \right|}^q}} {\rm{d}}x \ge \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{a}{2}{\left\| u \right\|^2} + \frac{b}{4}{\left\| u \right\|^4} - \frac{1}{{6{S^3}}}{\left\| u \right\|^6} - \beta _k^q\frac{\mu }{q}{\left\| u \right\|^q} \ge \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{a}{4}{\left\| u \right\|^2} - \beta _k^q\frac{\mu }{q}{\left\| u \right\|^q} \end{array} $ (16)

    $ {\rho _k} = {\left( {\frac{{4\mu \beta _k^q}}{{aq}}} \right)^{\frac{1}{{2 - q}}}} $,有ρk→0(k→∞).所以,存在k0,当kk0时,使得ρkR.因此,当uZk,‖u‖=ρkR时,有φμ(u)≥0.故条件(B1)成立.

    对于条件(B2),由于dim Yk<∞,所以Yk上的任意范数等价,则存在常数C4C5>0,有:

    $ \begin{array}{*{20}{c}} {{{\left| u \right|}_6} \le C_4^{\frac{1}{6}}\left\| u \right\|}&{{{\left| u \right|}_q} \le C_5^{\frac{1}{q}}\left\| u \right\|} \end{array} $

    那么对∀uYk,且‖u‖=γk,有

    $ \begin{array}{l} {\varphi _\mu }\left( u \right) \le \frac{a}{2}{\left\| u \right\|^2} + \frac{b}{4}{\left\| u \right\|^4} - \frac{{{C_4}}}{6}{\left\| u \right\|^6} - \frac{{{C_5}\mu }}{q}{\left\| u \right\|^q} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{a}{2}\gamma _k^2 + \frac{b}{4}\gamma _k^4 - \frac{{{C_4}}}{6}\gamma _k^6 - \frac{{{C_5}\mu }}{q}\gamma _k^q \end{array} $

    由于μ>0,C5>0,显然,存在充分小的γk,使得φμ(u)<0,所以条件(B2)成立.

    对于条件(B3),由(16)式,得

    $ {\varphi _\mu }\left( u \right) \ge - \beta _k^q\frac{\mu }{q}{\left\| u \right\|^q} $

    又由于βk→0(k→∞),存在k0,当kk0,且uZk,‖u‖≤ρk时,有$ {\varphi _\mu }(u) \ge - \beta _k^q\frac{\mu }{q}\rho _k^q $.故条件(B3)成立.由引理1知,存在μ*>0,使得对每个0<μμ*c<0,泛函φμ满足局部(PS)c*条件.定理1证毕.

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    Multiplicity of Solutions for Kirchhoff Equation with Concave and Convex Nonlinearities
    WANG Ya-qi, OU Zeng-qi     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: In this paper, we study a class of Kirchhoff equation $ \left\{ \begin{array}{l} - \left( {a + b\int_\mathit{\Omega} {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u = {\left| u \right|^4}u + \mu {\left| u \right|^{q - 2}}u\;\;\;\;\;\;\;x \in \mathit{\Omega} \\ u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \mathit{\Omega} \end{array} \right. $ with concave and convex nonlinearities, where Ω$ \mathbb{R} $3 is a smooth bounded domain with a, b > 0, 1 < q < 2, μ > 0. By means of the concentration compactness principle and a dual fountain theorem, we obtain the multiplicity of solutions about this equation.
    Key words: Kirchhoff equation    concave and convex nonlinearities    the concentration compactness principle    dual fountain theorem    
    X