西南师范大学学报(自然科学版)   2020, Vol. 45 Issue (2): 11-19.  DOI: 10.13718/j.cnki.xsxb.2020.02.003
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  • 含临界指数项和双重奇异项的Kirchhoff型椭圆边值方程的正解    [PDF全文]
    张黔 1,2, 邓志颖 1     
    1. 重庆邮电大学 理学院 应用数学系, 重庆 400065;
    2. 重庆数联铭信科技有限公司, 重庆 401121
    摘要:讨论了一类含临界指数项和双重奇异项的Kirchhoff型椭圆边值方程.应用Lions集中紧性原理和Ekeland变分原理,证明了该方程在适当条件下正解的存在性与多重性,推广和改进了一些最近的结果.
    关键词Kirchhoff型方程    Sobolev临界指数    奇异项    集中紧性原理    正解    

    讨论一类含有Sobolev临界指数项和双重奇异项的Kirchhoff型椭圆边值方程

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

    其中Ω$\mathbb{R}$N(N≥3)中包含原点的有界光滑区域,b>0,λ>0,0 < γ < 1均为常数,$0 \le \mu < \bar \mu = {\left( {\frac{{N - 2}}{2}} \right)^2}, {2^*} = \frac{{2N}}{{N - 2}}$是Sobolev嵌入临界指数,f:Ω$\mathbb{R}$L2(Ω)中给定的非负非平凡函数.

    形如(1)式的这类方程在理论探索和实际运用中具有重要意义,最早由文献[1]提出了数学模型

    $ {u_u} - \left( {a + b\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u = f\left( {x,u} \right) $ (2)

    这是一类经典的自由振动的弹性D′Alembert方程.在方程(2)中,u代表位移,f(xu)代表外力作用,b代表初始张力,a与弦的固有性质有关.文献[2]提出了关于方程(2)的一个抽象框架,对Kirchhoff型方程引入了泛函分析的研究方式,在此之后,学者们广泛应用变分方法研究方程

    $ \left\{ {\begin{array}{*{20}{l}} { - \left( {a + b\int_\mathit{\Omega } {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u = f\left( {x,u} \right)}&{x \in \mathit{\Omega }}\\ {u = 0}&{x \in \partial \mathit{\Omega }} \end{array}} \right. $ (3)

    特别是当方程(3)中的非线性项f(xu)含有Sobolev临界指数项时,方程(3)引起了学者们的广泛关注.文献[3]应用Ekeland变分原理、山路定理等变分方法得到了方程(3)多解的存在性;文献[4]应用Nehari流形和Ljusternik-Schnirelmann畴数理论得到了方程(3)多解的存在性.然而,关于含奇异项和临界指数项的Kirchhoff型椭圆方程的结果很少,文献[5]研究了这类问题,应用扰动方法、临界点理论得到了两个不同的正解.文献[6]讨论了一类含奇异项和临界指数项的p-Kirchhoff型椭圆边值方程

    $ \left\{ {\begin{array}{*{20}{l}} { - M{{\left\| u \right\|}^2}{\Delta _p}u = \lambda {u^{{p^ * } - 1}} + \rho \left( x \right){u^{ - \gamma }}}&{x \in \mathit{\Omega }}\\ {u = 0}&{x \in \partial \mathit{\Omega }} \end{array}} \right. $ (4)

    其中$\mathit{\Omega } \subset $$\mathbb{R}$N$M(s) = a + b{s^m}, a > 0, b > 0, {\Delta _p}u = {\mathop{\rm div}\nolimits} \left( {|\nabla u{|^{p - 2}}\nabla u} \right)$p-Laplace算子,作者应用Lions集中紧性原理、Vitali定理和Ekeland变分原理得到了方程(4)的两个正解.关于这类Kirchhoff型方程更进一步的研究参见文献[7-9].

    受文献[5-6]的启发,本文很自然地提出一个问题:对于方程(1),是否还存在正解以及多个正解?据我们所知,目前人们尚未研究过.解决方程(1)将面临3个方面的困难:首先,方程(1)含有非局部项$\left( {\int_\Omega | \nabla u{|^2}{\rm{d}}x} \right)\Delta u$,使得方程本身不再点点恒等;同时,嵌入$W_0^{1.2}(\Omega ){L^{{2^*}}}(\Omega )$是非紧的;最后,负指数项f(x)uγ导致能量泛函不可微,从而不能直接应用临界点理论处理这类问题.本文应用Lions集中紧性原理、Brézis-Lieb引理、Vitali定理和Ekeland变分原理克服了上述困难.本文得到的主要结果概括为:

    定理1  设Ω$\mathbb{R}$N是包含原点的有界光滑区域,0 < γ < 1,fΩ$\mathbb{R}$L2(Ω)中给定的非负非平凡函数,则存在${\lambda ^*}, \mathit{\Theta } > 0$,使得当$|f{|_2}\mathit{\Theta },\lambda \in \left( {0,{\lambda ^*}} \right)$时,方程(1)至少存在两个正解.

    注1  当a=1,b=0,μ=0时,方程(1)转化为半线性奇异椭圆边值方程,文献[10]应用Lions集中紧性原理和Ekeland变分原理得到了该方程的两个正解.

    1 方程(1)正解的存在性

    本节主要目的是证明方程(1)正解的存在性.首先,设p>1,定义$W_0^{1 \cdot p}(\mathit{\Omega } )$${\mathit{L}^\mathit{p}}(\mathit{\Omega })$中的范数分别为$||u|| = {\left( {\int_\mathit{\Omega } | \nabla u{|^\rho }{\rm{d}}x} \right)^{\frac{1}{\rho }}}, |u{|_p} = {\left( {\int_\mathit{\Omega } | u{|^\rho }{\rm{d}}x} \right)^{\frac{1}{\rho }}}$.讨论方程(1)的出发点是下述Hardy不等式[11]

    $ \int_\mathit{\Omega } {{{\left| x \right|}^{ - 2}}{{\left| u \right|}^2}{\rm{d}}x \le {{\left\| u \right\|}^2}/\bar \mu } \;\;\;\;\;\;\forall u \in W_0^{1,2}\left( \mathit{\Omega } \right) $

    $0 \le \mu < \bar \mu $时,在H01(Ω)中可定义范数

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

    由此可知,‖uμ与通常的范数‖u‖等价.定义嵌入W01,2(Ω)↺ L2*(Ω)的最佳临界常数为

    $ {S_\mu } = \mathop {\inf }\limits_{u \in H_0^1\left( \mathit{\Omega } \right)\backslash \left\{ 0 \right\}} \frac{{\left\| u \right\|_\mu ^2}}{{\left| u \right|_{{2^ * }}^2}} $ (5)

    由文献[12]可知,SμΩ无关.当Ω$\mathbb{R}$N时,Sμ不可达到.

    定义方程(1)所对应的能量泛函为IλH01(Ω)→$\mathbb{R}$

    $ {I_\lambda }\left( u \right) = \frac{1}{2}\left\| u \right\|_\mu ^2 + \frac{b}{4}{\left\| u \right\|^4} - \frac{\lambda }{{{2^ * }}}\left| u \right|_{{2^ * }}^{{2^ * }} - \frac{1}{{1 - \gamma }}\int_\mathit{\Omega } f \left( x \right){\left| u \right|^{1 - \gamma }}{\rm{d}}x $

    注意到,方程(1)中的奇异项f(x)uγ会导致泛函IλH01(Ω)上不可微,即IλC1(H01(Ω),$\mathbb{R}$).通常,对∀vH01(Ω),方程(1)的弱解uH01(Ω)满足

    $ \int_\mathit{\Omega } {\left( {\nabla u\nabla v - \mu {{\left| x \right|}^{ - 2}}uv} \right){\rm{d}}x} + b{\left\| u \right\|^2}\int_\mathit{\Omega } \nabla u\nabla v{\rm{d}}x - \lambda \int_\mathit{\Omega } {{u^{{2^ * } - 1}}} v{\rm{d}}x - \int_\mathit{\Omega } f \left( x \right){\left| u \right|^{ - \gamma }}v{\rm{d}}x = 0 $ (6)

    由(6)式,定义集合

    $ \mathit{\Lambda } = \left\{ {u \in H_0^1\left( \mathit{\Omega } \right):\left\| u \right\|_\mu ^2 + b{{\left\| u \right\|}^4} - \lambda \left| u \right|_{{2^*}}^{{2^*}} - \int_\Omega f (x){{\left| u \right|}^{1 - \gamma }}{\rm{d}}x = 0} \right\} $

    考虑

    $ {\varphi _u}\left( t \right) = \frac{{{t^2}}}{2}\left\| u \right\|_\mu ^2 + \frac{{b{t^4}}}{4}{\left\| u \right\|^4} - \frac{{\lambda {t^2}}}{{{2^ * }}}\left| u \right|_{{2^ * }}^{{2^ * }} - \frac{{{t^{1 - \gamma }}}}{{1 - \gamma }}\int_\mathit{\Omega } f \left( x \right){\left| u \right|^{1 - \gamma }}{\rm{d}}x $

    从而可得

    $ {{\varphi ''}_u}\left( 1 \right) = \left( {1 + \gamma } \right)\left\| u \right\|_\mu ^2 + b\left( {3 + \gamma } \right){\left\| u \right\|^4} - \lambda \left( {{2^ * } - 1 + \gamma } \right)\left| u \right|_{{2^ * }}^{{2^ * }} $

    可将Λ拆分成3部分,分别为Λ的局部极小值点、拐点和局部极大值点的集合:

    $ \begin{array}{*{20}{c}} {{\mathit{\Lambda }^ + } = \left\{ {u \in \mathit{\Lambda }:{{\varphi ''}_u}\left( 1 \right) > 0} \right\}}&{{\mathit{\Lambda }^0} = \left\{ {u \in \mathit{\Lambda }:{{\varphi ''}_u}\left( 1 \right) = 0} \right\}}&{{\mathit{\Lambda }^ - } = \left\{ {u \in \mathit{\Lambda }:{{\varphi ''}_u}\left( 1 \right) < 0} \right\}} \end{array} $

    在全文中,若无特别说明,CC0C1C2,…均表示正常数,“→”和“$\rightharpoonup$”分别表示相应空间中的强收敛和弱收敛,Br(x)表示以x为球心,r为半径的开球.为证明方程(1)正解的存在性,给出下述引理:

    引理1  泛函IΛH01(Ω)中有局部极小值m,且m < 0.

    证由(5)式和Hölder不等式可得

    $ \int_\mathit{\Omega } {{{\left| u \right|}^{{2^*}}}{\rm{d}}x} \le S_\mu ^{ - \frac{{{2^*}}}{2}}\left\| u \right\|_\mu ^{{2^*}} $ (7)
    $ \int_\mathit{\Omega } {f\left( x \right){{\left| u \right|}^{1 - \gamma }}{\rm{d}}x} \le {\left| f \right|_2}{\left| \mathit{\Omega } \right|^{\frac{{{2^ * } - 2 + 2\gamma }}{{2 \cdot {2^ * }}}}}\left| u \right|_{{2^*}}^{1 - \gamma } \le {C_0}{\left| f \right|_2}\left| u \right|_{{2^*}}^{1 - \gamma } \le {C_1}{\left| f \right|_2}\left\| u \right\|_\mu ^{1 - \gamma } $ (8)

    $ {I_\lambda }\left( u \right) = \frac{1}{2}\left\| u \right\|_\mu ^2 + \frac{b}{4}\left\| u \right\|_\mu ^4 - {C_2}{\left| f \right|_2}\left\| u \right\|_\mu ^{1 - \gamma } - \lambda {C_3}\left\| u \right\|_\mu ^{{2^ * }}\;\;\;\;\forall u \in H_0^1\left( \mathit{\Omega } \right) $ (9)

    由1-γ < 2及(9)式可得,存在λ1>0,对∀λ∈(0,λ1),存在ρR>0,使得IλSR={uH01(Ω):‖uμ=R}上Iλ(u)≥ρ,并且IλBR={uH01(Ω):‖uμR}上下方有界.从而对固定的λ∈(0,λ1),可以定义$m = \mathop {\inf }\limits_{u \in {B_R}} {I_\lambda }(u)$.由于0 < 1-γ < 1,从而由(9)式可知,对∀ω≠0及充分小的t>0,有${I_\lambda }(t\omega ) < 0$.从而可得m < 0.

    引理2  存在u1BR,使得Iλ(u1)=m.

      由引理1,对∀λ∈(0,λ1),有

    $ \left\{ \begin{array}{l} \frac{1}{2}\left\| u \right\|_\mu ^2 + \frac{b}{4}{\left\| u \right\|^4} - \frac{\lambda }{{{2^*}}}\left| u \right|_{{2^ * }}^{{2^ * }} \ge \rho \quad \forall u \in {S_R}\\ \frac{1}{2}\left\| u \right\|_\mu ^2 + \frac{b}{4}{\left\| u \right\|^4} - \frac{\lambda }{{{2^*}}}\left| u \right|_{{2^ * }}^{{2^ * }} \ge 0\quad \forall u \in {B_R} \end{array} \right. $ (10)

    m的定义可知,存在极小化序列{uk}⊂BR,使得Iλ(uk)→m < 0(k→∞).显然Iλ(uk)=Iλ(|uk|),从而可设uk≥0.又因为‖uμR,由有界集的弱紧性,存在子序列(仍记为{uk}),当k→∞时满足uk$\rightharpoonup$u1(xH01(Ω)),uk$\rightharpoonup$u1(xLs(Ω)(1≤s < 6)),uk(x)→u1(x)(a.e. xΩ).由(7)式知,ukL2*(Ω)中有界.又因H01(Ω)是自反空间,且BR是闭凸的,所以u1BR.再由Vitali定理可得

    $ \mathop {\lim }\limits_{k \to \infty } \int_\mathit{\Omega } f \left( x \right){\left| {{u_k}} \right|^{1 - \gamma }}{\rm{d}}x = \int_\mathit{\Omega } f \left( x \right){\left| {{u^1}} \right|^{1 - \gamma }}{\rm{d}}x $ (11)

    Ω=k=uku1,则由Brézis-Lieb引理,有

    $ {\left\| {{u_k}} \right\|^2} = {\left\| {{\omega _k}} \right\|^2} + {\left\| {{u^1}} \right\|^2} + o\left( 1 \right) $ (12)
    $ {\left\| {{u_k}} \right\|^4} = {\left\| {{\omega _k}} \right\|^4} + {\left\| {{u^1}} \right\|^4} + 2{\left\| {{\omega _k}} \right\|^2}{\left\| {{u^1}} \right\|^2} + o\left( 1 \right) $ (13)
    $ \left| {{u_k}} \right|_{{2^*}}^{{2^*}} = \left| {{\omega _k}} \right|_{{2^*}}^{{2^*}} + \left| {{u^1}} \right|_{{2^*}}^{{2^*}} + o\left( 1 \right) $ (14)

    注意到$\left\|u^{1}\right\|_{\mu}^{2}=\left\|u^{1}\right\|^{2}-\mu\left|\frac{u^{1}}{x}\right|_{2}^{2}$,故

    $ \left\| {{u_k}} \right\|_\mu ^2 = \left\| {{\omega _k}} \right\|_\mu ^2 + \left\| {{u^1}} \right\|_\mu ^2 + o\left( 1 \right) $ (15)

    u1=0,则ω=k=u=k,那么ω=kBR;若u1≠0,由(12)式知,当k充分大时有ω=kBR.因此,由(10)式得

    $ \frac{1}{2}\left\| {{\omega _k}} \right\|_\mu ^2 + \frac{b}{4}\left\| {{\omega _k}} \right\|_\mu ^4 - \frac{\lambda }{{{2^*}}}\left| {{\omega _k}} \right|_{{2^*}}^{{2^*}} \ge 0 $ (16)

    从而由(11)-(16)式可得

    $ \begin{array}{l} m = {I_\lambda }\left( {{u^1}} \right) + o\left( 1 \right) = {I_\lambda }\left( {{u^1}} \right) + \frac{1}{2}\left\| {{\omega _k}} \right\|_\mu ^2 + \frac{b}{4}\left\| {{\omega _k}} \right\|_\mu ^4 + \frac{b}{2}{\left\| {{\omega _k}} \right\|^2}{\left\| {{u^1}} \right\|^2} - \frac{\lambda }{{{2^*}}}\left| {{\omega _k}} \right|_{{2^*}}^{{2^*}} + o\left( 1 \right) \ge \\ \;\;\;\;\;\;{I_\lambda }\left( {{u^1}} \right) + \frac{b}{2}{\left\| {{\omega _k}} \right\|^2}{\left\| {{u^1}} \right\|^2} + o\left( 1 \right) \ge {I_\lambda }\left( {{u^1}} \right) + o\left( 1 \right) \end{array} $

    k→∞,可知mIλ(u1).再由m的定义,可得mIλ(u1).故Iλ(u1)=m < 0,且u1不恒等于0,故u1Iλ的局部极小解.

    定理1正解存在性的证明  显然,对∀λ∈(0,λ1),引理1和引理2都成立.现只需证明u1是方程(1)的正弱解.由引理2可知

    $ \begin{array}{*{20}{c}} {\mathop {\min }\limits_{t \in \mathbb{R}} {I_\lambda }\left( {{u^1} + t\varphi } \right) = {{\left. {{I_\lambda }\left( {{u^1} + t\varphi } \right)} \right|}_{t = 0}} = {I_\lambda }\left( {{u^1}} \right)}&{\forall \varphi \in H_0^1\left( \mathit{\Omega } \right)} \end{array} $

    因此

    $ 0 = \int_\mathit{\Omega } {\left( {\nabla {u^1}\nabla \varphi - \mu {{\left| x \right|}^{ - 2}}{u^1}\varphi } \right){\rm{d}}x} + b{\left\| {{u^1}} \right\|^2}\int_\mathit{\Omega } \nabla {u^1}\nabla \varphi {\rm{d}}x - \lambda \int_\mathit{\Omega } {{{\left( {{u^1}} \right)}^{{2^ * } - 1}}} \varphi {\rm{d}}x - \int_\mathit{\Omega } f (x){\left| {{u^1}} \right|^{ - \gamma }}\varphi {\rm{d}}x $

    其中∀φH01(Ω).由此可知u1是方程(1)的弱解.对∀φH01(Ω),u1≥0,φ≥0,t>0,可得0≤Iλ(u1+)-Iλ(u1).易知

    $ \begin{array}{*{20}{c}} {\frac{1}{2}\left\| {{u^1} + t\varphi } \right\|_\mu ^2 - \frac{1}{2}\left\| {{u^1}} \right\|_\mu ^2 + \frac{b}{4}{{\left\| {{u^1} + t\varphi } \right\|}^4} - \frac{b}{4}{{\left\| {{u^1}} \right\|}^4} \ge 0}&{\forall \varphi \in H_0^1\left( \mathit{\Omega } \right),\varphi \ge 0} \end{array} $ (17)

    将(17)式两边同时除以t(t>0),再令t→0,可得u1H01(Ω),且

    $ - \left( {1 + b{{\left\| {{u^1}} \right\|}^2}} \right)\Delta {u^1} - \mu {\left| x \right|^{ - 2}}{u^1} \ge 0 $

    由强极大值原理知u1>0(xΩ).

    2 方程(1)正解的多重性

    本节主要证明方程(1)正解的多重性,主要在Λ上讨论方程(1)的正解,并由Lions集中紧性原理、Vitali定理和Ekeland变分原理等方法证明得到.

    引理3   IΛΛ上强制.

      对∀uΛ,有

    $ \left\| u \right\|_\mu ^2 + b{\left\| u \right\|^4} - \lambda \left| u \right|_{{2^*}}^{{2^*}} - \int_\mathit{\Omega } f \left( x \right){\left| u \right|^{1 - \gamma }}{\rm{d}}x = 0 $

    从而可得

    $ {I_\lambda }\left( u \right) \ge \left( {\frac{1}{2} - \frac{1}{{{2^ * }}}} \right)\left\| u \right\|_\mu ^2 + \left( {\frac{1}{4} - \frac{1}{{{2^ * }}}} \right)b\left\| u \right\|_\mu ^4 - \left( {\frac{1}{{1 - \gamma }} - \frac{1}{{{2^ * }}}} \right){C_1}{\left| f \right|_2}\left\| u \right\|_\mu ^{1 - \gamma } $

    由于0 < 1-γ < 2,则$\mathop {\lim }\limits_{u{_\mu } \to \infty } {I_\lambda }(u) = + \infty $,即IλΛ上强制,且下方有界.

    引理4  存在Θ>0,λ2>0,使得当λ∈(0,λ2),0 < |f|2Θ时,有Λ≠∅,Λ0={0},且ΛH01(Ω)中是闭集.

      设$\varphi (t) = {t^{1 + \gamma }}u_\mu ^2 + b{t^{3 + \gamma }}{\left\| u \right\|^4} - \lambda {t^{{2^*} - 1 + \gamma }}|u|_{{2^*}}^{{2^*}}$.令tε满足φ(tε)=0,则有

    $ {t_\varepsilon } > {t_0} = {\left[ {\frac{{\left( {1 + \gamma } \right)\left\| u \right\|_\mu ^2}}{{\lambda \left( {{2^ * } - 1 + \gamma } \right)\left| u \right|_{{2^*}}^{{2^*}}}}} \right]^{\frac{1}{{{2^ * } - 2}}}} $

    注意到:对∀0 < t < tε,有φ(t)>0;对∀t>tε,有φ′(t) < 0.那么φ在0 < t < tε上递增,在t>tε上递减.于是φ(t)可在tε点处取得最大值.因此,取λ充分小,推断可得

    $ {\varphi _{\max }} = \varphi \left( {{t_\varepsilon }} \right) > \varphi \left( {{t_0}} \right) > \left( {\frac{{{2^*} - 2}}{{{2^*} - 1 + \gamma }}} \right){\left( {\frac{{1 + \gamma }}{{{2^*} - 1 + \gamma }}} \right)^{\frac{{1 + \gamma }}{{{2^*} - 2}}}}S_\mu ^{\frac{{{2^*} - 1 + \gamma }}{{{2^*} - 2}}}\left| u \right|_{{2^*}}^{1 - \gamma } $

    再由(7)式和(8)式有

    $ {\varphi _{\max }} - \int_\mathit{\Omega } f \left( x \right){\left| u \right|^{1 - \gamma }}{\rm{d}}x > 0 $ (18)

    那么对∀0 < |f|2Θ,有(18)式成立,其中

    $ \mathit{\Theta } = {\left| \mathit{\Omega } \right|^{\frac{{{2^*} - 2 + 2\gamma }}{{2 \cdot {2^*}}}}}\left( {\frac{{{2^*} - 2}}{{{2^*} - 1 + \gamma }}} \right){\left( {\frac{{1 + \gamma }}{{{2^*} - 1 + \gamma }}} \right)^{\frac{{1 + \gamma }}{{{2^*} - 2}}}}S_\mu ^{\frac{{{2^*} - 1 + \gamma }}{{{2^*} - 2}}} $

    因此,存在唯一的正数$t^{-}=t^{-}(u)>t_{\varepsilon}$,使得$\varphi \left( {{t^ - }} \right) = \int_\mathit{\Omega } f (x)|u{|^{1 - \gamma }}{\rm{d}}x, {\varphi ^\prime }\left( {{t^ - }} \right) < 0$,这就意味着t-uΛ-,即对∀0 < |f|2Θ,有Λ.

    用反证法证明Λ0={0}.假设uΛ0\{0},则

    $ \begin{array}{*{20}{c}} {0 = \left( {1 + \gamma } \right)\left\| u \right\|_\mu ^2 + b\left( {3 + \gamma } \right){{\left\| u \right\|}^4} - \lambda \left( {{2^*} - 1 + \gamma } \right)\left| u \right|_{{2^*}}^{{2^*}} = }\\ {\left\| u \right\|_\mu ^2 + b{{\left\| u \right\|}^4} - \lambda \left| u \right|_{{2^*}}^{{2^*}} - \int_\mathit{\Omega } f \left( x \right){{\left| u \right|}^{1 - \gamma }}{\rm{d}}x} \end{array} $ (19)

    因此

    $ \frac{{{2^*} - 2}}{{{2^*} - 1 + \gamma }}\left\| u \right\|_\mu ^2 + \frac{{{2^*} - 4}}{{{2^*} - 1 + \gamma }}b\left\| u \right\|_\mu ^4 - \int_\mathit{\Omega } f (x){\left| u \right|^{1 - \gamma }}{\rm{d}}x = 0 $ (20)

    由(7),(19)式及Young不等式可得

    $ 2\sqrt {b(1 + \gamma )(3 + \gamma )} \left\| u \right\|_\mu ^3 \le \lambda \left( {{2^*} - 1 + \gamma } \right)S_\mu ^{ - \frac{{{2^ * }}}{2}}\left\| u \right\|_\mu ^{{2^ * }} $

    $ {\left\| u \right\|_\mu } \ge {\left\{ {\frac{{2\sqrt {b(1 + \gamma )(3 + \gamma )} }}{{\lambda \left( {{2^*} - 1 + \gamma } \right)}}S_\mu ^{\frac{{{2^ * }}}{2}}} \right\}^{\frac{1}{{{2^*} - 3}}}} $ (21)

    由(8),(20)式及Young不等式可得

    $ \frac{{2\sqrt {\left( {{2^ * } - 2} \right)\left( {{2^ * } - 4} \right)b} }}{{{2^ * } - 1 + \gamma }}\left\| u \right\|_\mu ^3 \le \int_\mathit{\Omega } f \left( x \right){\left| u \right|^{1 - \gamma }}{\rm{d}}x \le {C_1}{\left| f \right|_2}\left\| u \right\|_\mu ^{1 - \gamma } $

    由于0 < 1-γ < 1,则

    $ {\left\| u \right\|_\mu } \le {\left\{ {\frac{{{C_1}{{\left| f \right|}_2}\left( {{2^ * } - 1 + \gamma } \right)}}{{2\sqrt {\left( {{2^ * } - 2} \right)\left( {{2^ * } - 4} \right)b} }}} \right\}^{\frac{1}{{2 + \gamma }}}} $ (22)

    $ \lambda < {\lambda _3} = \frac{{2\sqrt {b\left( {1 + \gamma } \right)\left( {3 + \gamma } \right)} }}{{{2^ * } - 1 + \gamma }}{S^{\frac{{{2^ * }}}{{{\mu ^2}}}}} \cdot {\left\{ {\frac{{{C_1}{{\left| f \right|}_2}\left( {{2^*} - 1 + \gamma } \right)}}{{2\sqrt {\left( {{2^*} - 2} \right)\left( {{2^*} - 4} \right)b} }}} \right\}^{\frac{{3 - {2^*}}}{{2 + \gamma }}}} $

    时,(22)式与(21)式矛盾.综上所述,对∀λ∈(0,λ2),有Λ0={0}.

    下证ΛH01(Ω)中是闭集.令{un}⊂Λ使得unu(xH01(Ω)),存在子序列(仍记为{un}),使得unu(a.e. xΩ),且$\mathop {\lim }\limits_{n \to \infty } {\left| {{u_n}} \right|_2} \cdot = |u{|_2}$.再由Λ的定义可得

    $ \mathop {\lim }\limits_{n \to \infty } \left\{ {\left( {1 + \gamma } \right)\left\| {{u_n}} \right\|_\mu ^2 + b\left( {3 + \gamma } \right){{\left\| {{u_n}} \right\|}^4} - \lambda \left( {{2^*} - 1 + \gamma } \right)\left| u \right|_{{2^ * }}^{{2^ * }}} \right\} \le 0 $ (23)

    由(23)式可知uΛ0Λ.若Λ不是闭集,则uΛ0,此时u≡0.对∀{un}⊂Λ

    $ {\left| {{u_n}} \right|_{{2^ * }}} > {\left\{ {\frac{{2\sqrt {b\left( {1 + \gamma } \right)\left( {3 + \gamma } \right)} }}{{\lambda \left( {{2^ * } - 1 + \gamma } \right)}}} \right\}^{\frac{1}{{{2^ * } - 3}}}}S_\mu ^{\frac{3}{{2 \cdot {2^ * } - 6}}} $

    从而,当n→∞时,有

    $ \mathop {\lim }\limits_{n \to \infty } {\left| {{u_n}} \right|_{{2^ * }}} \ge {\left\{ {\frac{{2\sqrt {b\left( {1 + \gamma } \right)\left( {3 + \gamma } \right)} }}{{\lambda \left( {{2^ * } - 1 + \gamma } \right)}}} \right\}^{\frac{1}{{{2^ * } - 3}}}}S_\mu ^{\frac{3}{{2 \cdot {2^ * } - 6}}} > 0 $ (24)

    (24) 式与u=0矛盾.因此对∀λ∈(0,λ2),有uΛ,且ΛH01(Ω)中是闭集.

    引理5  存在λ3>0,使得对∀uΛλ∈(0,λ3),有Iλ(u)≥0.

      用反证法.假设存在$\bar u \in {\mathit{\Lambda }^ - }$,使得$I_{\lambda}(\bar{u})<0$,即

    $ \frac{1}{2}\left\| {\bar u} \right\|_\mu ^2 + \frac{b}{4}{\left\| {\bar u} \right\|^4} - \frac{\lambda }{{{2^*}}}\left| {\bar u} \right|_2^{{2^*}} - \frac{1}{{1 - \gamma }}\int_\mathit{\Omega } f \left( x \right){\left| {\bar u} \right|^{1 - \gamma }}{\rm{d}}x < 0 $

    uΛΛ,则

    $ \left( {\frac{1}{2} - \frac{1}{{{2^ * }}}} \right)\left\| {\bar u} \right\|_\mu ^2 + \left( {\frac{1}{4} - \frac{1}{{{2^*}}}} \right)b{\left\| {\bar u} \right\|^4} - \left( {\frac{1}{{1 - \gamma }} - \frac{1}{{{2^*}}}} \right)\int_\mathit{\Omega } f \left( x \right){\left| {\bar u} \right|^{1 - \gamma }}{\rm{d}}x < 0 $

    再由(7),(8)式及Young不等式,可得

    $ \sqrt {\frac{{b\left( {{2^*} - 2} \right)\left( {{2^*} - 4} \right)}}{{2{{\left( {{2^*}} \right)}^2}}}} {\left( {S_\mu ^3\left| {\bar u} \right|_{{2^ * }}^{{2^ * }}} \right)^{\frac{1}{2}}} < \left( {\frac{1}{{1 - \gamma }} - \frac{1}{{{2^*}}}} \right){C_0}{\left| f \right|_2}\left| {\bar u} \right|_{{2^ * }}^{1 - \gamma } $

    $ {\left| {\bar u} \right|_{{2^ * }}} < {\left\{ {\frac{{\left( {{2^*} - 1 + \gamma } \right){C_0}{{\left| f \right|}_2}}}{{\sqrt {\frac{{b\left( {{2^*} - 2} \right)\left( {{2^*} - 4} \right)}}{2}} }}\left( {1 - \gamma } \right)S_\mu ^{\frac{3}{2}}} \right\}^{\frac{1}{{2 + \gamma }}}} $

    $ {\lambda _3} = {\left\{ {\frac{{\sqrt {\frac{{b\left( {{2^*} - 2} \right)\left( {{2^*} - 4} \right)}}{2}} \left( {1 - \gamma } \right)S_\mu ^{\frac{3}{2}}}}{{{C_0}{{\left| f \right|}_2}\left( {{2^*} - 1 + \gamma } \right)}}} \right\}^{\frac{{{2^*} - 3}}{{2 + \gamma }}}}\left\{ {\frac{{2\sqrt {b(1 + \gamma )(3 + \gamma )} S_\mu ^{\frac{3}{2}}}}{{{2^*} - 1 + \gamma }}} \right\} $

    则对任意的λ < λ3,有

    $ {\left| {\bar u} \right|_{{2^ * }}} < {\left\{ {\frac{{2\sqrt {b(1 + \gamma )(3 + \gamma )} }}{{\lambda \left( {{2^*} - 1 + \gamma } \right)}}} \right\}^{\frac{1}{{{2^ * } - 3}}}}S_\mu ^{\frac{2}{{2 \cdot {2^ * } - 6}}} $

    这与(24)式矛盾.因此,引理5得证.

    引理6  若uΛ,则存在ε>0及可微函数f=f(ω)>0,其中ωH01(Ω),‖ω‖ < ε,使得

    $ f\left( 0 \right) = 1\;\;\;f\left( \omega \right)\left( {u + \omega } \right) \in {\mathit{\Lambda }^ - },\forall \omega \in H_0^1\left( \mathit{\Omega } \right) $

      定义F$\mathbb{R}$×H01(Ω) $\longrightarrow$$\mathbb{R}$,有

    $ F\left( {t,\omega } \right) = {t^{1 + \gamma }}\left\| {u + \omega } \right\|_\mu ^2{\rm{d}}x + b{t^{3 + \gamma }}{\left\| {u + \omega } \right\|^4} - \lambda {t^{{2^*} - 1 + \gamma }}\left| {u + \omega } \right|_{{2^ * }}^{{2^ * }}: - \int_\mathit{\Omega } f \left( x \right){\left| {u + \omega } \right|^{1 - \gamma }}{\rm{d}}x $

    由于uΛΛ,可知F(1,0)=0,且

    $ {F_t}\left( {1,0} \right) = \left( {1 + \gamma } \right)\left\| u \right\|_\mu ^2 + b\left( {3 + \gamma } \right){\left\| u \right\|^4} - \lambda \left( {{2^*} - 1 + \gamma } \right)\left| u \right|_{{2^ * }}^{{2^ * }}: < 0 $

    由隐函数定理,存在$\bar{\varepsilon}$>0及可微函数f=f(ω)>0,使得f(0)=1,f(ω)(u+ω)∈Λ,∀ωH01(Ω),‖ω‖ < $\bar{\varepsilon}$.显然,给定充分小的ε>0(ε < $\bar{\varepsilon}$),满足

    $ f\left( \omega \right)\left( {u + \omega } \right) \in {\mathit{\Lambda }^ - }\;\;\;\forall \omega \in H_0^1\left( \mathit{\Omega } \right),\left\| \omega \right\| < \varepsilon $

    引理7  对∀λ>0,方程(1)在H01(Ω)中有弱解u2.

      由$0 < |f{|_2}\mathit{\Theta }$及引理4可知${\mathit{\Lambda }^ - } \ne \emptyset ,{m_ - } = \mathop {\inf }\limits_{u \in {\Delta ^ - }} {I_\lambda }(u) > - \infty $.由Ekeland变分原理,存在一个极小化序列{vn}⊂Λ,满足

    $ \begin{array}{*{20}{c}} {{I_\lambda }\left( {{v_n}} \right) < {m_ - } + \frac{1}{n}}&{{I_\lambda }\left( {{v_n}} \right) \le {I_\lambda }\left( v \right) + \frac{1}{n}\left\| {v - {v_n}} \right\|}&{\forall v \in {\mathit{\Lambda }^ - }} \end{array} $

    Iλ(vn)=Iλ(|vn|),可设vn≥0在Ω中,且{vn}收敛到一个函数(有必要时收敛到子序列),记作u2≥0,且满足:vn$\rightharpoonup$u2(xH01(Ω)),vnu2(a.e. xΩ).令u=vnΛΩ=φH01(Ω),t>0充分小,可得可微函数fn(t)=fn(),满足fn(0)=1,fn(t)(vn+)∈Λ.由于ΛΛ,可得

    $ f_n^2\left( t \right)\left\| {{v_n} + t\varphi } \right\|_\mu ^2 + bf_n^4\left( t \right){\left\| {{v_n} + t\varphi } \right\|^4} - \lambda f_n^{{2^*}}\left( t \right)\left| {{v_n} + t\varphi } \right|_{{2^ * }}^{{2^ * }} - f_n^{1 - \gamma }\left( t \right)\int_\mathit{\Omega } f \left( x \right){\left| {{v_n} + t\varphi } \right|^{1 - \gamma }}{\rm{d}}x = 0 $
    $ \left\| {{v_n}} \right\|_\mu ^2 + b{\left\| {{v_n}} \right\|^4} - \lambda \left| {{v_n}} \right|_{{2^ * }}^{{2^ * }} - \int_\mathit{\Omega } f (x){\left| {{v_n}} \right|^{1 - \gamma }}{\rm{d}}x = 0 $

    由Ekeland变分原理可得

    $ \begin{array}{*{20}{c}} {\frac{1}{n}\left[ {\left| {{f_n}(t) - 1} \right|{{\left\| {{v_n}} \right\|}_\mu } + t{f_n}(t){{\left\| \varphi \right\|}_\mu }} \right] \ge }\\ {\frac{1}{n}{{\left\| {{f_n}\left( t \right)\left( {{v_n} + t\varphi } \right) - {v_n}} \right\|}_\mu } \ge {I_\lambda }\left( {{v_n}} \right) - {I_\lambda }\left[ {{f_n}\left( t \right)\left( {{v_n} + t\varphi } \right)} \right] = }\\ {\frac{{1 - f_n^2\left( t \right)}}{2}\left\| {{v_n}} \right\|_\mu ^2 + b\frac{{1 - f_n^4\left( t \right)}}{4}{{\left\| {{v_n}} \right\|}^4} - \lambda \frac{{1 - f_n^{{2^ * }}(t)}}{{{2^*}}}\left| {{v_n} + t\varphi } \right|_{{2^ * }}^{{2^ * }} - \frac{{1 - f_n^{1 - \gamma }(t)}}{{1 - \gamma }}\int_\mathit{\Omega } f (x){{\left| {{v_n} + t\varphi } \right|}^{1 - \gamma }}{\rm{d}}x + }\\ {\frac{{f_n^2\left( t \right)}}{2}\left( {\left\| {{v_n}} \right\|_\mu ^2 - \left\| {{v_n} + t\varphi } \right\|_\mu ^2} \right) + \frac{{bf_n^4\left( t \right)}}{4}\left( {{{\left\| {{v_n}} \right\|}^4} - {{\left\| {{v_n} + t\varphi } \right\|}^4}} \right) - \frac{\lambda }{{{2^*}}}\left( {\left| {{v_n}} \right|_{{2^ * }}^{{2^ * }} - \left| {{v_n} + t\varphi } \right|_{{2^ * }}^{{2^ * }}} \right) - }\\ {\frac{1}{{1 - \gamma }}\left[ {\int_\mathit{\Omega } f (x){{\left| {{v_n}} \right|}^{1 - \gamma }}{\rm{d}}x - \int_\mathit{\Omega } f (x){{\left| {{v_n} + t\varphi } \right|}^{1 - \gamma }}{\rm{d}}x} \right]} \end{array} $ (25)

    将(25)式两边同时除以t(t>0),再令t→0,由此可得

    $ \begin{array}{l} \int_\mathit{\Omega } f (x){\left| {{v_n}} \right|^{1 - \gamma }}\varphi {\rm{d}}x \le \\ \frac{1}{n}\left[ {\left| {{{f'}_n}\left( 0 \right)} \right|{{\left\| {{v_n}} \right\|}_\mu } + {{\left\| \varphi \right\|}_\mu }} \right] + \int_\mathit{\Omega } {\left( {\nabla {v_n}\nabla \varphi - \mu {{\left| x \right|}^{ - 2}}{v_n}\varphi } \right){\rm{d}}x} + b{\left\| {{v_n}} \right\|^2}\int_\mathit{\Omega } \nabla {v_n}\nabla \varphi {\rm{d}}x - \lambda \int_\mathit{\Omega } {v_n^2} { \cdot ^{ - 1}}\varphi {\rm{d}}x \end{array} $

    再由文献[13],存在C4>0使得|fn(0)|≤C4.当n→∞时,由Fatou引理可得

    $ \begin{array}{l} \int_\mathit{\Omega } f (x){\left( {{u^2}} \right)^{ - \gamma }}\varphi {\rm{d}}x \le \mathop {\lim \inf }\limits_{n \to \infty } \int_\mathit{\Omega } f (x){\left| {{v_n}} \right|^{ - \gamma }}\varphi {\rm{d}}x \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_\mathit{\Omega } {\left( {\nabla {u^2}\nabla \varphi - \mu {{\left| x \right|}^{ - 2}}{u^2}\varphi } \right){\rm{d}}x} + b\left\| {{u^2}} \right\|\int_\mathit{\Omega } \nabla {u^2}\nabla \varphi {\rm{d}}x - \lambda \int_\mathit{\Omega } {{{\left( {{u^2}} \right)}^{{2^ * } - 1}}} \varphi {\rm{d}}x \end{array} $

    由于φ的任意性,该不等式对-φ也成立,因此可得

    $ \int_\mathit{\Omega } {\left( {\nabla {u^2}\nabla \varphi - \mu {{\left| x \right|}^{ - 2}}{u^2}\varphi } \right){\rm{d}}x} + b{\left\| {{u^2}} \right\|^2}\int_\mathit{\Omega } \nabla {u^2}\nabla \varphi {\rm{d}}x - \lambda \int_\mathit{\Omega } {{{\left( {{u^2}} \right)}^{{2^ * } - 1}}\varphi {\rm{d}}x} - \int_2 f (x){\left( {{u^2}} \right)^{ - \gamma }}\varphi {\rm{d}}x = 0 $

    这意味着u2是方程(1)的弱解.

    引理8  存在λ4>0,使得当0 < λ < λ4时,有u2Λ.

      对∀uΛΛ,可知

    $ {I_\lambda }\left( u \right) < \frac{1}{{4\left( {1 - \gamma } \right)}}\left[ {\left( {1 + \gamma } \right)\left\| u \right\|_\mu ^2 + b\left( {3 + \gamma } \right){{\left\| u \right\|}^4}} \right] + \frac{{\lambda \left( {{2^ * } - 1 + \gamma } \right)}}{{{2^ * }\left( {1 - \gamma } \right)}}\left| u \right|_{{2^ * }}^{{2^ * }}: = \frac{{\lambda \left( {{2^*} + 4} \right)\left( {{2^*} - 1 + \gamma } \right)}}{{4 \cdot {2^*}\left( {1 - \gamma } \right)}}\left| u \right|_{{2^ * }}^{{2^ * }} $

    易知,取$\lambda<\tilde{\lambda}_{4}$,当${\tilde \lambda _4}$满足

    $ \frac{{\lambda \left( {{2^*} + 4} \right)\left( {{2^*} - 1 + \gamma } \right)}}{{4 \cdot {2^*}\left( {1 - \gamma } \right)}}\left| u \right|_{{2^ * }}^{{2^ * }} < \frac{1}{N}S_\mu ^{\frac{N}{2}} $

    时,我们有$m_{-}<\frac{1}{N} S_{\mu}^{\frac{N}{2}}$.

    下证u2Λ,由于Λ是闭集,且vn$\rightharpoonup$u2(xH01(Ω)),现只需证‖vnμ→‖u2μ.由Lions集中紧性原理[14],存在非负有界测度ηνν,使得|▽vn|2$\rightharpoonup$η,|vn|2*$\rightharpoonup$ν,|x|-2|vn|2$\rightharpoonup$ν在测度意义下成立,并存在至多可数集$\mathscr{J} $,点集${\left\{ {{x_j}} \right\}_{j \in \mathscr{J}}} \subset \mathit{\bar \Omega }\backslash \{ 0\} , \;\;\;{\kern 1pt} {\left\{ {{\eta _j} \ge 0} \right\}_{j \in \mathscr{J} \cup (0)}}, \;\;\;{\kern 1pt} {\left\{ {{\nu _j} \ge 0} \right\}_{j \in \mathscr{J} \cup \{ 0\} }}, \;\;\;{\kern 1pt} {\tilde \nu _0} \ge 0$,使得:

    ${\rm{ (a) }}\eta \ge {\left| {\nabla {u^2}} \right|^2} + \sum\limits_{j \in \mathscr{J}} {{\eta _j}} {\delta _{{x_j}}} + {\eta _0}{\delta _0}$,

    $({\rm{b}})\nu = {\rm{ }}{\left| {{u^2}} \right|^{{2^*}}} + \sum\limits_{j \in \mathscr{J}} {{\nu _j}} {\delta _{{x_j}}} + {\nu _0}{\delta _0}$,

    ${\rm{ (c) }}\tilde \nu = |x{|^{ - 2}}{\left| {{v_n}} \right|^2} + {\tilde \nu _0}{\delta _0}$,

    $({\rm{d}})\;\;\;{\kern 1pt} {S_0}\nu _j^{\frac{2}{*}} \le {\eta _j}$,

    $({\rm{e}}){S_\mu }v_0^{\frac{2}{{{2^*}}}}{\eta _0} - \mu {\tilde v_0}$.

    其中δxj(j$\mathscr{J} $∪{0})是xj处的Dirac测度.令n→∞,可得

    $ {\left\| {{v_n}} \right\|^2} \to \int_\mathit{\Omega } {\rm{d}} \eta \ge {\left\| {{u^2}} \right\|^2} + \sum\limits_{j \in \mathscr{J}} {{S_0}} \nu _j^{\frac{2}{{{2^*}}}} + {\eta _0} $ (26)
    $ \left| {{v_n}} \right|_{{2^*}}^{{2^*}} \to \int_\mathit{\Omega } {\rm{d}} \nu = \left| {{u^2}} \right|_{{2^*}}^{{2^*}} + \sum\limits_{j \in \mathscr{J}} {{\nu _j}} + {\nu _0} $ (27)
    $ {\int_\mathit{\Omega } {\left| x \right|} ^{ - 2}}{\left| {{v_n}} \right|^2}{\rm{d}}x \to \int_\mathit{\Omega } {\rm{d}} \tilde \nu = {\int_\mathit{\Omega } {\left| x \right|} ^{ - 2}}{\left| {{v_n}} \right|^2}{\rm{d}}x + {{\tilde \nu }_0} $ (28)

    再由Vitali定理,可得

    $ \mathop {\lim }\limits_{k \to \infty } \int_\mathit{\Omega } f (x){\left| {{u_n}} \right|^{1 - \gamma }}{\rm{d}}x = \int_\mathit{\Omega } f (x){\left| {{u^2}} \right|^{1 - \gamma }}{\rm{d}}x $ (29)

    xj≠0是测度ην的一个奇异点,因为对∀j$\mathscr{J} $,有xj≠0,选取充分小的ε>0,使得0∈Bxj(ε)且Bxi(ε)∩Bxj(ε)=,其中ijij$\mathscr{J} $.现定义ψε是以xj为中心的光滑截断函数,且0≤ψε≤1.当$\left| {x - {x_j}} \right|\frac{\varepsilon }{2}$时,ψε=1;当|xxj|≥ε时,ψε=0,并且有$\left|\nabla \psi_{\varepsilon}\right| \leqslant \frac{4}{\varepsilon}$.

    显然,类似于引理7的证明方法,对∀φH01(Ω),当n→∞时,有

    $ \int_\mathit{\Omega } {\left( {\nabla {v_n}\nabla \varphi - \mu \frac{{{v_n}\varphi }}{{{{\left| x \right|}^2}}}} \right){\rm{d}}x} + b{\left\| {{v_n}} \right\|^2}\int_\mathit{\Omega } \nabla {v_n}\nabla \varphi {\rm{d}}x - \lambda \int_\mathit{\Omega } {{{\left( {{v_n}} \right)}^{{2^ * } - 1}}\varphi {\rm{d}}x} - \int_\mathit{\Omega } {f\left( x \right)} {\left( {{v_n}} \right)^{ - \gamma }}\varphi {\rm{d}}x = o\left( 1 \right) $

    由于ψεvnH01(Ω),则

    $ \begin{array}{l} \int_\mathit{\Omega } {\left( {\nabla {v_n}\nabla \left( {{\psi _\varepsilon }{v_n}} \right) - \mu \frac{{{v_n}{\psi _\varepsilon }{v_n}}}{{{{\left| x \right|}^2}}}} \right){\rm{d}}x} + b{\left\| {{v_n}} \right\|^2}\int_\mathit{\Omega } \nabla {v_n}\nabla \left( {{\psi _\varepsilon }{v_n}} \right){\rm{d}}x\\ - \lambda \int_\mathit{\Omega } {{{\left( {{v_n}} \right)}^{{2^*} - 1}}} {\psi _\varepsilon }{v_n}{\rm{d}}x - \int_\mathit{\Omega } f (x){\left( {{v_n}} \right)^{ - \gamma }}{\psi _\varepsilon }{v_n}{\rm{d}}x = o\left( 1 \right) \end{array} $

    依据(26)-(29)式可推断出

    $ \begin{array}{l} - \left( {b\int_\mathit{\Omega } {\rm{d}} \eta + 1} \right)\int_\mathit{\Omega } {{\psi _\varepsilon }} {\rm{d}}\eta + \mu \int_\mathit{\Omega } {{\psi _\varepsilon }} {\rm{d}}\tilde \nu + \int_\mathit{\Omega } f (x){\left( {{u^2}} \right)^{1 - \gamma }}{\psi _\varepsilon }{\rm{d}}x + \lambda \int_\mathit{\Omega } {{\psi _\varepsilon }} {\rm{d}}\nu = \\ \left( {b\int_\mathit{\Omega } {\rm{d}} \eta + 1} \right)\int_\mathit{\Omega } {\left( {\nabla {v_n}\nabla {\psi _\varepsilon }} \right)} {v_n}{\rm{d}}x + o\left( 1 \right)\;\;\;\;n \to \infty \end{array} $

    n→∞时,有

    $ 0 = \mathop {\lim }\limits_{\varepsilon \to 0} \left[ {\left( {b\int_\mathit{\Omega } {\rm{d}} \eta + 1} \right)\int_\mathit{\Omega } {{\psi _\varepsilon }} {\rm{d}}\eta - \mu \int_\mathit{\Omega } {{\psi _\varepsilon }} {\rm{d}}\tilde \nu - \int_\mathit{\Omega } f (x){{\left( {{u^2}} \right)}^{1 - \gamma }}{\psi _\varepsilon }{\rm{d}}x - \lambda \int_\mathit{\Omega } {{\psi _\varepsilon }} {\rm{d}}\nu } \right] \ge {\eta _j} - \lambda {\nu _j} $ (30)

    由(30)式可得$\lambda {\nu _j} \ge {\eta _j},{\lambda _{{\nu _j}}} \ge {S_0}\nu _j^{\frac{2}{{{2^*}}}}$.从而再由(d)得出:要么(ⅰ) νj=0;要么(ⅱ) ${\nu _j} \ge {\left( {\frac{{{S_0}}}{\lambda }} \right)^{\frac{N}{2}}}$.同理,对于x=0的集中情形,由(e)可得:要么(ⅲ) ν0=0;要么(ⅳ) ${\nu _0} \ge {\left( {\frac{{{S_\mu }}}{\lambda }} \right)^{\frac{N}{2}}}$.

    下证情形(ⅱ)及(ⅳ)不成立.用反证法,假设存在j0,使得${\nu _{{j_0}}} \ge {\left( {\frac{{{S_0}}}{\lambda }} \right)^{\frac{N}{2}}}$.

    $ \begin{array}{l} {m_ - } = \mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{2}\left\| {{v_n}} \right\|_\mu ^2 + \frac{b}{4}{{\left\| {{v_n}} \right\|}^4} - \frac{\lambda }{{{2^*}}}\left| {{v_n}} \right|_{{2^ * }}^{{2^ * }} - \frac{1}{{1 - \gamma }}\int_\mathit{\Omega } f (x){{\left| {{v_n}} \right|}^{1 - \gamma }}{\rm{d}}x} \right) \ge \\ \;\;\;\;\;\;\;\frac{1}{2}\left( {{{\left\| {{u^2}} \right\|}^2} + \sum\limits_{j \in \mathscr{J} \cup \left\{ 0 \right\}} {{\eta _j}} - \mu \int_\mathit{\Omega } {\frac{{{{\left| {{u^2}} \right|}^2}}}{{|x{|^2}}}} {\rm{d}}x - \mu {{\tilde \nu }_0}} \right) - \frac{\lambda }{{{2^*}}}\left( {\left| {{u^2}} \right|_{{2^ * }}^{{2^ * }}: + \sum\limits_{j \in \mathscr{J} \cup \left\{ 0 \right\}} {{v_j}} } \right) - \frac{1}{{1 - \gamma }}\int_\mathit{\Omega } f (x){\left| {{u^2}} \right|^{1 - \gamma }}{\rm{d}}x \ge \\ \;\;\;\;\;\;\;{I_\lambda }\left( {{u^2}} \right) + \frac{1}{N}S_0^{\frac{N}{2}}{\lambda ^{1 - \frac{N}{2}}} > {I_\lambda }\left( {{u^2}} \right) + \frac{1}{N}S_0^{\frac{N}{2}} \end{array} $

    事实上,${m_ - } < \frac{1}{N}S_n^{\frac{N}{2}}$,由此可知Iλ(u2) < 0,故u2≠0,且

    $ 0 < \frac{b}{4}{\left\| {{u^2}} \right\|^4} < \frac{1}{2}\left\| {{u^2}} \right\|_\mu ^2 + \frac{b}{4}{\left\| {{u^2}} \right\|^4} < \frac{\lambda }{{{2^*}}}\left| {{u^2}} \right|_{{2^ * }}^{{2^ * }} + \frac{1}{{1 - \gamma }}\int_\mathit{\Omega } f \left( x \right){\left| {{u^2}} \right|^{1 - \gamma }}{\rm{d}}x $

    因此

    $ \begin{array}{l} {m_ - } = \mathop {\lim }\limits_{n \to \infty } {I_\lambda }\left( {{v_n}} \right) = \mathop {\lim }\limits_{n \to \infty } \left[ {{I_\lambda }\left( {{v_n}} \right) - \frac{1}{2}\left( {\left\| {{v_n}} \right\|_\mu ^2 + b{{\left\| {{v_n}} \right\|}^4} - \lambda \left| {{v_n}} \right|_{{2^ * }}^{{2^ * }} - \int_\mathit{\Omega } f (x){{\left| {{v_n}} \right|}^{1 - \gamma }}{\rm{d}}x} \right)} \right] \ge \\ \;\;\;\;\;\;\;\mathop {\lim }\limits_{n \to \infty } \left( {\frac{\lambda }{2}\left| {{u^2}} \right|_{{2^ * }}^{{2^ * }} + \frac{1}{2}\int_\mathit{\Omega } f (x){{\left| {{u^2}} \right|}^{1 - \gamma }}{\rm{d}}x} \right) \ge \frac{\lambda }{2}\left( {\left| {{u^2}} \right|_{{2^ * }}^{{2^ * }} + \sum\limits_{j \in J \cup \left\{ 0 \right\}} {{\nu _j}} } \right) + \frac{1}{2}\int_\mathit{\Omega } f (x){\left| {{u^2}} \right|^{1 - \gamma }}{\rm{d}}x \ge \\ \;\;\;\;\;\;\;\frac{\lambda }{2}\int_\mathit{\Omega } {{{\left| {{u^2}} \right|}^{{2^ * }}}} {\rm{d}}x + \frac{1}{2}{C_0}{\left| f \right|_2}\left| {{u^2}} \right|_{{2^*}}^{1 - \gamma } + \frac{1}{2}S_0^{\frac{N}{2}}{\lambda ^{1 - \frac{N}{2}}} > \frac{1}{N}S_0^{\frac{N}{2}} \end{array} $ (31)

    $\bar{\lambda}_{4}>0$,使得

    $ \frac{{{{\bar \lambda }_4}}}{2}\left| {{u^2}} \right|_{{2^*}}^{{2^*}} + \frac{1}{2}{C_0}{\left| f \right|_2}\left| {{u^2}} \right|_{{2^ * }}^{1 - \gamma } < 0 $

    对∀ $\lambda < {\bar \lambda _4}$,使得(31)式最后一个不等式成立.这与${m_ - } < \frac{1}{N}S_\mu ^{\frac{N}{2}}$矛盾.同理,对x=0的情形,(ⅳ)可得$m_{-}>\frac{1}{N} S_{\mu}^{\frac{N}{2}}$.因此$\lambda_{4}=\min \left\{\tilde{\lambda}_{4}, \bar{\lambda}_{4}\right\}$,那么对∀j$\mathscr{J} $∪{0}都有νj=0,则|vn|2*$\rightharpoonup$|u2|2*,且对∀λ∈(0,λ4),有uΛ.

    定理1正解多重性的证明  令λ=min{λi}(i=2,3,4).显然,对∀λ∈(0,λ),引理3-引理8都成立.现只需要证明u2>0(xΩ).由引理7和引理8可知:u2IΛΛ-中的极小解.类似方程(1)正解存在性的证明方法,可证得:u2>0(xΩ).综上所述,取λ*=min{λ1λ′},定理1得证.

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    Positive Solutions for Kirchhoff-Type Elliptic Boundary Value Equations with Critical Exponent and Double Singular Terms
    ZHANG Qian 1,2, DENG Zhi-ying 1     
    1. Department of Applied Mathematics, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;
    2. Chongqing Shulian Mingxin Technology Limited Company, Chongqing 401121, China
    Abstract: This paper deals with a class of Kirchhoff-type elliptic boundary value equations involving critical exponent and double singular terms.Based upon the Lions concentration compactness principle and the Ekeland variational principle, we obtain the existence and multiplicity of positive solutions for the problem under the appropriate conditions, some recent results are generalized and significantly improved.
    Key words: Kirchhoff-type equation    Sobolev critical exponent    singular terms    the concentration compactness principle    positive solution    
    X