中国学术期刊网 » 论文 » 理学论文 » 数学论文 » 一类Kirchhoff型p(x)-调和方程的多重解论文正文

一类Kirchhoff型p(x)-调和方程的多重解

中国学术期刊网【数学论文】 编辑:天问 山东大学学报(理学版) 2016-11-04一类Kirchhoff型p(x)-调和方程的多重解论文作者:张申贵,原文发表在《山东大学学报(理学版)杂志》,经中国学术期刊网小编精心整理,仅供您参考。

关键词: Kirchhoff型方程 p(x)-调和算子 Navier边值问题 临界点
摘要: 研究了一类Kirchhoff型p(x)-调和方程。利用临界点理论中的喷泉定理, 获得了多重解存在的充分条件, 推广和改进了一些已有的结果。

0 引言

带有p(x)-Laplace和p(x)-调和算子的微分方程来自于非线性弹性问题和流体力学, 该模型刻画了“逐点异性”的物理现象。文献[1]研究了非线性弹性力学中的变指数微分方程模型, 文献[2]讨论了变流体力学中的变指数变分问题, 文献[3]利用变指数微分方程模型研究了图像处理问题。近年来, 变分原理和临界点理论已用于研究p(x)-调和方程的可解性[4-10]。

文中, 考虑Kirchhoff型p(x)-调和方程Navier边值问题

$ \left\{ {\begin{array}{*{20}{l}} {\lef( {a + b\int_\mathit \Omega {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}} - \lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)\lef( {\Delta _{p\lef( x \right)}^2u - \lambda {{\left| u \right|}^{p\lef( x \right) - 2}}u} \right) = f\lef( {x,u} \right),}&{x \in \mathit \Omega ,}\\ {u = \Delta u = 0,}&{x \in \partial \mathit \Omega ,} \end{array}} \right. $ (0.1)

其中a, b为正常数, Ω是RN中具有光滑边界的有界区域, λ≤0, 称Δp(x)2u=Δ(|Δu|p(x)-2Δu)为p(x)-调和算子, 且$ 1 < {p^-}: = \mathop {\inf p}\limits_\mathit \Omega \lef( x \right) \le {p^ + }: = \mathop {\sup p\lef( x \right)}\limits_\mathit \Omega < + \infty $。问题(0.1)来源于物理学家G.Kirchhoff建立的基尔霍夫方程, 该模型的特点是带有非局部系数, 并且可用于描述生物的种群密度等平均量。

美国学者P.H.Rabinowitz给出著名的超线性条件:(AR)存在μ>2, L>0, 使得0 < μF(x, s)≤f(x, s)s, 对所有x∈Ω和|s|≥L成立。当a=0, b=1, 问题(0.1)退化为p(x)-调和方程。特别地, 当条件(AR)成立时, 文献[10]中得到了p(x)-调和方程Navier边值问题解的存在性定理。条件(AR)可以保证非线性项f(x, u)关于变量u在无穷远处是超线性的。(AR)条件被广泛地用于椭圆型偏微分方程边值问题、狄拉克方程、波方程、薛定谔方程和Hamilton系统解存在性的研究中, 但是很多超线性函数并不满足条件(AR)。

本文目的是研究Kirchhoff型p(x)-调和方程Navier边值问题(0.1)。方法为将系统(0.1)的解转化为定义在Orlic-Sobolev空间W01, p(x)(Ω)∩W2, p(x)(Ω)上一个泛函的临界点, 在比(AR)更弱的超线性条件下, 利用临界点理论中的喷泉定理得到此类问题无穷多高能量解存在的充分条件。

1 预备知识

记Lp(x)(Ω)={u|u是可测的实值函数, $ \int_\mathit \Omega {{{\left| {u\lef( x \right)} \right|}^{p\lef( x \right)}}{\rm{d}}x < \infty } $, 其范数为

$ {\left| u \right|_{p\lef( x \right)}} = \inf \left\{ {\lambda > 0:\int_\mathit \Omega {{{\left| {\frac{{u\lef( x \right)}}{\lambda }} \right|}^{p\lef( x \right)}}{\rm{d}}x \le 1} } \right\}。 $

记Wk, p(x)={u∈Lp(x):|Dαu∈Lp(x)(Ω), |α|≤k}, 其范数为$ {\left\| u \right\|_{{W^{k, p\lef( x \right)}}}} = \sum\limits_{\left| \alpha \right| \le k} {{{\left| {{D^\alpha }u} \right|}_{p\lef( x \right)}}} $其中α=(α1, …, αN), $ \left| \alpha \right|{\rm{ = }}\sum\limits_{i = 1}^N {{\alpha _i}}, $$ {D^\alpha }u = \frac{{{\partial ^{\left| \alpha \right|}}u}}{{{\partial ^{{\alpha _1}}}{x_1} \cdots {\partial ^{{\alpha _N}}}{x_N}}} $。

W0k, p(x)为C0∞(Ω)在Wk, p(x)中的闭包。W0k, p(x)(Ω)和Wk, p(x)(Ω)都是可分自反的Banach空间。记Sobolev空间X:=W2, p(x)(Ω)∩W01, p(x)(Ω), 其范数为

$ \left\| u \right\| = \inf \left\{ {\alpha > 0:\int_\mathit \Omega {\lef( {{{\left| {\frac{{\Delta u\lef( x \right)}}{\alpha }} \right|}^{p\lef( x \right)}}-\lambda {{\left| {\frac{{u\lef( x \right)}}{\alpha }} \right|}^{p\lef( x \right)}}} \right){\rm{d}}x \ge 1} } \right\}。 $

引理1.1[10]若q(x)∈C+(Ω), 满足q(x) < p*(x), ∀x∈Ω, 则X:=W2, p(x)(Ω)∩W01, p(x)(Ω)$ \hookrightarrow $Lq(x)(x)是紧嵌入。

在X上定义能量泛函φ如下:

$ \varphi \lef( u \right) = a\int_\mathit \Omega {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}}-\lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x + \frac{b}{2}{{\lef( {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}}-\lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} \right)}^2}-\int_\mathit \Omega {F\lef( {x, u} \right)} } {\rm{d}}x, $

其中$ F\lef( {x, u} \right) = \int_0^u {f\lef( {x, t} \right){\rm{d}}t}, $且φ∈C1(X, R),

$ \begin{array}{*{20}{l}} {\left\langle {\varphi '\lef( u \right), v} \right\rangle = \lef( {a + b\int_\mathit \Omega {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}}-\lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)\int_\mathit \Omega {\lef( {{{\left| {\Delta u} \right|}^{p\lef( x \right)-2}}\Delta u\Delta v-\lambda {{\left| {\Delta u} \right|}^{p\lef( x \right) - 2}}uv} \right){\rm{d}}x - } }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_\mathit \Omega {v\lef( {x, u} \right)v{\rm{d}}x。} } \end{array} $

则u∈X是问题(0.1)的解等价于u是泛函φ的临界点。

引理1.2[10]记$ J\lef( u \right) = \int_\mathit \Omega {\lef( {{{\left| {\Delta u} \right|}^{p\lef( x \right)}}-\lambda {{\left| u \right|}^{p\lef( x \right)}}} \right){\rm{d}}x}, $$ \forall u \in W_0^{1, p\lef( x \right)}\lef( \Omega \right) $。则下列结论成立:

(ⅰ)$ \left\| u \right\| < \lef( { = ; > } \right)1 \Leftrightarrow J\lef( u \right) < \lef( { = ; > } \right)1; $
(ⅱ)$ \left\| u \right\| \le 1 \Rightarrow {\left\| u \right\|^{{p^ + }}} \le J\lef( u \right) \le {\left\| u \right\|^{{p^-}}}; $

(ⅲ)$ \left\| u \right\| \ge 1 \Rightarrow {\left\| u \right\|^{{p^-}}} \le J\lef( u \right) \le {\left\| u \right\|^{{p^ + }}}; $

(ⅳ)$ \left\| {{u_n}} \right\| \to 0 \Leftrightarrow J\lef( {{u_n}} \right) \to 0;\left\| {{u_n}} \right\| \to \infty \Leftrightarrow J\lef( {{u_n}} \right) \to \infty ;\forall {u_n} \in W_0^{1, p\lef( x \right)}\lef( \Omega \right) $。

令X:=W01, p(x)(Ω)∩W2, p(x)(Ω), X可分自反的Banach空间, 则存在{ej}⊂X, 使得

$ X = {\rm{span}}\left\{ {{e_j}\left| {j \in N} \right.} \right\}, {X_j} = {\rm{span}}\left\{ {{e_j}} \right\}, {Y_k} = \oplus _{j = 1}^k{X_j}, {Z_k} = \overline {{ \oplus _{j \ge k}}{X_j}}。 $

定义1.1设X为Banach空间, 若泛函φ∈C1(X, R)满足:对任何点列{un}⊂X, 由{φ(un)}有界, (1+‖un‖)‖φ′(un)‖→0(n→∞), 蕴含{un}有收敛子列, 则称泛函φ满足(C)条件。

引理1.3[11] (喷泉定理)若φ∈C1(X, R), 满足:φ(0)=0, φ(-u)=φ(u), 且

(ⅰ)对充分大的k∈N, 存在rk > 0, 当k→∞时, 有bk:=infu∈Zk, ‖u‖=rkφ(u)→+∞;

(ⅱ)对ρk > rk > 0, 有ak:=maxu∈Yk, ‖u‖=ρkφ(u)≤0;

(ⅲ)泛函φ满足(C)条件,

则泛函φ有一列趋向于+∞的临界值。

2 主要结果

假设以下条件成立:

(F0)设f(x, u)∈C(Ω×R, R), 存在c>0, α(x)∈C+(Ω), 使得

$ \left| {f\lef( {x, s} \right)} \right| \le c\lef( {1 + {{\left| s \right|}^{\alpha \lef( x \right)-1}}} \right), $

对所有x∈Ω和s∈R成立。其中C+(Ω)={h|h∈C(Ω), h(x) > 1, ∀x∈Ω}, p(x) < α(x) < p*(x), 当2p(x) < N时, $ p*\lef( x \right) = \frac{{Np\lef( x \right)}}{{N-2p\lef( x \right)}} $; 当2p(x)≥N时, p*(x)=+∞。

(F1) F(x, u)=-K(x, u)+W(x, u), 其中K, W:Ω×R→R是C1映照。

(F2) F(x, u)关于u是偶函数, 且∫ΩF(x, 0)dx=0。

(K1)存在正常数k1, k2, 使得k1|u|2p-≤K(x, u)≤k2|u|2p+, 对(x, u)∈Ω×R成立。

(K2) K(x, u)≤ku(x, u)u≤2p+K(x, u), 对(x, u)∈Ω×R成立。

(W1)设$ \mathop {\lim }\limits_{\left| u \right| \to \infty } \frac{{W\lef( {x, u} \right)}}{{{{\left| u \right|}^{2{p^ + }}}}} = + \infty $, 对x∈Ω一致成立。

(W2)存在常数L > 0, C1 > 0, 当|u|≥L时, 有$ \frac{1}{{2{p^ + }}}{W_u}\lef( {x, u} \right)u-W\lef( {x, u} \right) \ge {C_1}{\left| u \right|^{{p^-}}} $。

(W3)存在常数L > 0, C2 > 0, 当|u|≥L时, 有

$ \left| {{W_u}{{\lef( {x, u} \right)}^\sigma }} \right| \le {C_2}{\left| s \right|^{\sigma \lef( {{p^- }- 1} \right)}}\left[{\frac{1}{{2{p^ + }}}{W_u}\lef( {x, u} \right)u-W\lef( {x, u} \right)} \right], $

其中$ \sigma > \frac{{p*}}{{\lef( {1-\theta } \right)\lef( {p*-{p^-}} \right)}}, \theta \in \lef( {0, 1} \right) $。

定理2.1设(F0)~(F2), (K1)~(K2), (W1)~(W3)成立, 则问题‘(0.1)有一列解{uk}k∈N满足:当k→∞时,

$ a\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_k}} \right|}^{p\lef( x \right)}}-\lambda {{\left| {{u_k}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x + \frac{b}{2}{{\lef( {\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_k}} \right|}^{p\lef( x \right)}}-\lambda {{\left| {{u_k}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)}^2}-\int_\mathit \Omega {F\lef( {x, {u_k}} \right)} } {\rm{d}}x \to + \infty 。 $

注1文献[9]中定理1.1对应于本文定理2.1中a=0, b=1, K=0的特殊情形。当a=0, b=1, p(x)=2时, 则p+=p-=2。由条件(AR)可以推出条件(F1)~(F3)成立[12]。另一方面, K(x, u)=(1+e-|x|)u4, $ W\lef( {x, u} \right) = \lef( {2-\frac{1}{{1 + \left| x \right|}}} \right){u^4}\ln \lef( {1 + {u^4}} \right) $, 则F, K, W满足定理1中的条件, 但不满足条件(AR)。

证明第1步证明泛函φ满足(C)条件, 设{un}⊂W2, p(x)(Ω)∩W01, p(x)(Ω), 使得{φ(un)}有界, (1+‖un‖)‖φ′(un)‖→0(n→∞), 则存在C3 > 0使得

$ \left\| {\varphi \lef( {{u_n}} \right)} \right\| \le {C_3},\lef( {1 + \left\| {{u_n}} \right\|} \right)\left\| {\varphi '\lef( {{u_n}} \right)} \right\| \le {C_3} $ (2.1)

对一切自然数n成立。利用条件(F0)、(W2), 存在常数C4 > 0, 使得

$ \frac{1}{{2{p^ + }}}{W_u}\lef( {x,s} \right)s - W\lef( {x,s} \right) \ge {C_1}{\left| s \right|^{{p^ - }}} - {C_4} $ (2.2)

对所有x∈Ω和u∈R成立。由λ≤0, $ {p^ + }: = \mathop {\sup p}\limits_\mathit \Omega \lef( x \right) \ge p\lef( x \right) $, 有

$ \begin{array}{*{20}{l}} {a\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x + \frac{b}{2}{{\lef( {\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)}^2} \ge } }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{2{p^ + }}}\lef( {a + b\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)\int_\mathit \Omega {{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}{\rm{d}}x} ,} \end{array} $ (2.3)

由式(2.2)、(2.3)和条件(K2)、(F1), 并注意到$ {p^ + }: = \mathop {\sup p}\limits_\mathit \Omega \lef( x \right) \ge p\lef( x \right) $, 有

$ \begin{array}{*{20}{l}} {\frac{{\lef( {2{p^ + } + 1} \right){C_3}}}{{2{p^ + }}} \ge \varphi \lef( {{u_n}} \right) - \frac{1}{{2{p^{\rm{ + }}}}}\left\langle {\varphi '\lef( {{u_n}} \right),{u_n}} \right\rangle = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x + \frac{b}{2}{{\lef( {\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)}^2} - } }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{2{p^ + }}}\lef( {a + b\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)\int_\mathit \Omega {{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}{\rm{d}}x} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{2{p^ + }}}\int_\mathit \Omega {f\lef( {x,{u_n}} \right){u_n}{\rm{d}}x} - \int_\mathit \Omega {F\lef( {x,{u_n}} \right){\rm{d}}x} \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{2{p^ + }}}\int_\mathit \Omega {f\lef( {x,{u_n}} \right){u_n}{\rm{d}}x} - \int_\mathit \Omega {F\lef( {x,{u_n}} \right){\rm{d}}x} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{2{p^ + }}}\int_\mathit \Omega {{W_u}\lef( {x,{u_n}} \right){u_n}{\rm{d}}x} - \int_\mathit \Omega {W\lef( {x,{u_n}} \right){\rm{d}}x} - \frac{1}{{2{p^ + }}}\int_\mathit \Omega {{K_u}\lef( {x,{u_n}} \right){u_n}{\rm{d}}x} \\+ \frac{1}{{2{p^ + }}}\int_\mathit \Omega {K\lef( {x,{u_n}} \right){\rm{d}}x} \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{2{p^ + }}}\int_\mathit \Omega {{W_u}\lef( {x,{u_n}} \right){u_n}{\rm{d}}x} - \int_\mathit \Omega {W\lef( {x,{u_n}} \right){\rm{d}}x} \ge {C_1}\int_\mathit \Omega {{{\left| {{u_n}} \right|}^{{p^ - }}}{\rm{d}}x - {C_4}\left| \mathit \Omega \right|。} } \end{array} $ (2.4)

反设{un}在W2, p(x)(Ω)∩W01, p(x)(Ω)中无界, 当n→∞时, $ {v_n} = \frac{{{u_n}}}{{\left\| {{u_n}} \right\|}} $, ‖vn‖=1。由式(2.4), 有

$ \int_\mathit \Omega {{{\left| {{v_n}} \right|}^{{p^ - }}}{\rm{d}}x} = \frac{1}{{{{\left\| u \right\|}^{{p^ - }}}}}\int_\mathit \Omega {{{\left| {{u_n}} \right|}^{{p^ - }}}{\rm{d}}x} \le \frac{{{C_5}}}{{{{\left\| u \right\|}^{{p^ - }}}}} \to 0,\lef( {n \to \infty } \right)。 $ (2.5)

因为$ \sigma > \frac{{p*}}{{\lef( {1-\theta } \right)\lef( {p*-{p^-}} \right)}}, \theta \in \lef( {0, 1} \right), p* > {p^ - } $, 有$ \sigma > 1, \frac{{\sigma-1}}{{\theta \sigma }} > 1, \frac{{\lef( {1-\theta } \right)\sigma {p^-}}}{{\lef( {1 - \theta } \right)\sigma - 1}} < p* $。

利用Sobolev嵌入定理(引理1.1), $ X \Leftrightarrow {L^{\frac{{\lef( {1-\theta } \right)\sigma {p^-}}}{{\lef( {1-\theta } \right)\sigma p - 1}}}}\lef( \mathit \Omega \right) $为紧嵌入, 则

$ \int_\mathit \Omega {{{\left| {{v_n}} \right|}^{\frac{{\lef( {1 - \theta } \right)\sigma {p^ - }}}{{\lef( {1 - \theta } \right)\sigma - 1}}}}{\rm{d}}x} \ge {C_6}{\left\| u \right\|^{\frac{{\lef( {1 - \theta } \right)\sigma - 1}}{{\lef( {1 - \theta } \right)\sigma {p^ - }}}}} = {C_6}。 $ (2.6)

由HÖlder不等式, 并注意到$ \frac{1}{\sigma } + \frac{1}{{\sigma '}} = 1 $, 由式(2.5)、(2.6), 有

$ \begin{array}{*{20}{l}} {\int_\mathit \Omega {{{\left| {{v_n}} \right|}^{{p^ - }\sigma '}}{\rm{d}}x} = \int_\mathit \Omega {{{\left| {{v_n}} \right|}^{\frac{{\sigma {p^ - }}}{{\sigma - 1}}}}{\rm{d}}x} = \int_\mathit \Omega {{{\left| {{v_n}} \right|}^{\frac{{\theta \sigma {p^ - }}}{{\sigma - 1}}}}{{\left| {{v_n}} \right|}^{\frac{{\lef( {1 - \theta } \right)\sigma {p^ - }}}{{\sigma - 1}}}}{\rm{d}}x} \le }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left[ {\int_\mathit \Omega {{{\lef( {{{\left| {{v_n}} \right|}^{\frac{{\theta \sigma {p^ - }}}{{\sigma - 1}}}}} \right)}^{\frac{{\sigma - 1}}{{\theta \sigma }}}}{\rm{d}}x} } \right]}^{\frac{{\theta \sigma }}{{\sigma - 1}}}}{{\left[ {\int_\mathit \Omega {{{\lef( {{{\left| {{v_n}} \right|}^{\frac{{\lef( {1 - \theta } \right)\sigma {p^ - }}}{{\sigma - 1}}}}} \right)}^{\frac{1}{{1 - \frac{{\theta \sigma }}{{\sigma - 1}}}}}}{\rm{d}}x} } \right]}^{1 - \frac{{\theta \sigma }}{{\sigma - 1}}}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\lef( {\int_\mathit \Omega {{{\left| {{v_n}} \right|}^{{p^ - }}}{\rm{d}}x} } \right)}^{\frac{{\theta \sigma }}{{\sigma - 1}}}}{{\lef( {\int_\mathit \Omega {{{\left| {{v_n}} \right|}^{\frac{{\lef( {1 - \theta } \right)\sigma {p^ - }}}{{\lef( {1 - \theta } \right)\sigma - 1}}}}{\rm{d}}x} } \right)}^{1 - \frac{{\theta \sigma }}{{\sigma - 1}}}} \le }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\lef( {\int_\mathit \Omega {{{\left| {{v_n}} \right|}^{{p^ - }}}{\rm{d}}x} } \right)}^{\frac{{\theta \sigma }}{{\sigma - 1}}}}{{\lef( {{C_6}} \right)}^{1 - \frac{{\theta \sigma }}{{\sigma - 1}}}} \to 0,\lef( {n \to \infty } \right)。} \end{array} $ (2.7)

当n充分大时, 由条件(F3)和式(2.1)、(2.4), 有

$ \begin{array}{*{20}{l}} {\int_\mathit \Omega {{{\lef( {\frac{{\left| {{W_u}\lef( {x,{u_n}} \right)} \right|}}{{{{\left| {{u_n}} \right|}^{{p^ - } - 1}}}}} \right)}^\sigma }{\rm{d}}x \le {C_2}\int_\mathit \Omega {\left[ {\frac{1}{{2{p^ + }}}{W_u}\lef( {x,{u_n}} \right){u_n} - W\lef( {x,{u_n}} \right)} \right]} {\rm{d}}x \le } }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {C_2}\left[ {\varphi \lef( {{u_n}} \right) - \frac{{\left\langle {\varphi '\lef( {{u_n}} \right),{u_n}} \right\rangle }}{{2{p^ + }}}} \right] \le \frac{{\lef( {2{p^ + } + 1} \right){C_2}{C_3}}}{{2{p^ + }}},} \end{array} $ (2.8)

注意到$ \frac{1}{\sigma } + \frac{1}{{\sigma '}} = 1 $, $ {v_n} = \frac{{{u_n}}}{{\left\| {{u_n}} \right\|}} $, 利用HÖlder不等式, 式(2.7)、(2.8), 当n→∞, 有

$ {\left[ {\frac{{\lef( {2{p^ + } + 1} \right){C_2}{C_3}}}{{2{p^ + }}}} \right]^{\frac{1}{\sigma }}}{\lef( {\int_\mathit \Omega {{{\left| {{v_n}} \right|}^{{p^ - }\sigma '}}{\rm{d}}x} } \right)^{\frac{1}{{\sigma '}}}} \to 0。 $ (2.9)

当n→∞时, 由‖un‖→∞, 可取‖un‖≥1, 由引理1.2中(ⅲ), 条件(K1)、(K2), 有

$ \begin{array}{*{20}{l}} {\left\langle {\varphi '\lef( {{u_n}} \right), {u_n}} \right\rangle = \lef( {a + b\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}}-\lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)\int_\mathit \Omega {\lef( {{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}}-\lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}} \right){\rm{d}}x + } }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_\mathit \Omega {{K_u}\lef( {x, {u_n}} \right){u_n}{\rm{d}}x}-\int_\mathit \Omega {{W_u}\lef( {x, {u_n}} \right){u_n}{\rm{d}}x} \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a\int_\mathit \Omega {\lef( {{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}} - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}} \right){\rm{d}}x + } {k_1}\int_\mathit \Omega {{{\left| {{u_n}} \right|}^{2{p^ - }}}{\rm{d}}x - } \int_\mathit \Omega {{W_u}\lef( {x, {u_n}} \right){u_n}{\rm{d}}x} \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left\| {{u_n}} \right\|}^{{p^ - }}}\lef( {1 - \int_\mathit \Omega {\frac{{{W_u}\lef( {x, {u_n}} \right){u_n}}}{{{{\left\| {{u_n}} \right\|}^{{p^ - }}}}}{\rm{d}}x} } \right).} \end{array} $

由式(2.1), 当n充分大时, 有

$ a - \int_\mathit \Omega {\frac{{{W_u}\lef( {x,{u_n}} \right){u_n}}}{{{{\left\| {{u_n}} \right\|}^{{p^ - }}}}}} {\rm{d}}x \le o\lef( 1 \right)。 $ (2.10)

由式(2.9)、(2.10), 有a≤o(1), 这与常数a>0矛盾。故{un}在X:=W2, p(x)(Ω)∩W01, p(x)(Ω)中有界。注意到X是自反空间, 则存在u∈X, 使得{un}在X中弱收敛于u, 且{un}在Lα(x)(Ω)中强收敛于u。由HÖlder不等式, 条件(F0)和$ \frac{1}{{\alpha \lef( x \right)}} + \frac{1}{{\alpha '\lef( x \right)}} = 1 $, 有

$ \begin{array}{*{20}{c}} {\left| {\int_\mathit \Omega {\lef( {x, {u_n}} \right)} \lef( {{u_n}-u} \right){\rm{d}}x} \right| \le \int_\mathit \Omega {\left| {f\lef( {x, {u_n}} \right)} \right|\left| {{u_n}-u} \right|{\rm{d}}x} \le r\int_\mathit \Omega {\left| {1 + {{\left| {{u_n}} \right|}^{\alpha \lef( x \right)-1}}} \right|\left| {{u_n} - u} \right|{\rm{d}}x} \le }\\ {2r{{\left| {{u_n}} \right|}^{\alpha \lef( x \right) - 1}}\left| {_{\lef( {\alpha '\lef( x \right)} \right)}{{\left| {{u_n} - u} \right|}_{\alpha \lef( x \right)}}} \right. \to 0, \lef( {n \to \infty } \right)。} \end{array} $

当n→∞时, 有φ′(un)→0, 并注意到$ a + b\int_\mathit \Omega {\frac{{{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right)}}-\lambda {{\left| {{u_n}} \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} > a > 0 $, 当n→∞时, 有

$ \int_\mathit \Omega {\left[ {{{\left| {\Delta {u_n}} \right|}^{p\lef( x \right) - 2}}\Delta {u_n}\Delta \lef( {{u_n} - u} \right) - \lambda {{\left| {{u_n}} \right|}^{p\lef( x \right) - 2}}{u_n}\lef( {{u_n} - u} \right)} \right]} {\rm{d}}x \to 0。 $ (2.11)

定义$ \left\langle {Au, v} \right\rangle = \int_\mathit \Omega {\left[{{{\left| {\Delta u} \right|}^{p\lef( x \right)-2}}\Delta u\Delta v-\lambda {{\left| u \right|}^{p\lef( x \right)-2}}uv} \right]} {\rm{d}}x \to 0 $。由文献[10]知, 连续映射A:X→X*具有性质(S+), 即{un}在X中弱收敛于u, 且$ \mathop {\lim {\kern 1pt} {\kern 1pt} {\kern 1pt} sup}\limits_{n \to \infty } \left\langle {A\lef( {{u_n}} \right)-A\lef( u \right), {u_n}-u} \right\rangle \le 0 $, 可推出{un}在X中强收敛于u。由式(2.11), $ \mathop {\lim {\kern 1pt} {\kern 1pt} {\kern 1pt} sup}\limits_{n \to \infty } \left\langle {A\lef( {{u_n}} \right)-A\lef( u \right), {u_n}-u} \right\rangle = 0 $, 则{un}在X中强收敛于u。

第2步验证泛函φ满足引理1.3(喷泉定理)中条件(ⅰ)。

由条件(F0), p(x) < α(x) < p*(x), 有

$ \left| {F\lef( {x,s} \right)} \right| \le {C_7}{\left| s \right|^{\alpha \lef( x \right)}} + {C_8}\left| s \right| $ (2.12)

对所有x∈Ω和s∈R成立。

当|u|α(x)≤1时, 则 $ \int_\mathit \Omega {{{\left| u \right|}^{\alpha \lef( x \right)}}{\rm{d}}x} \le \left| u \right|_{\alpha \lef( x \right)}^{{\alpha ^ - }} $; 当|u|α(x) > 1时, 则 $ \int_\mathit \Omega {{{\left| u \right|}^{\alpha \lef( x \right)}}{\rm{d}}x} \le \left| u \right|_{\alpha \lef( x \right)}^{{\alpha ^ - }} \le {\lef( {{\beta _k}\left\| u \right\|} \right)^{{\alpha ^ + }}} $。令βk=sup{|u|α(x)|‖u‖=1, u∈Zk}, 其中|u|α(x)表示Lα(x)(Ω)中的范数。当α(x) < p*(x)时, 有$ \mathop {\lim }\limits_{k \to \infty } {\beta _k} = 0 $[10]。

对u∈Zk, 使得‖u‖=rk>1, 由(2.12)及引理1.2中(ⅲ), 有

$ \begin{array}{*{20}{l}} {\varphi \lef( u \right) = a\int_\mathit \Omega {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}}- \lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x + \frac{b}{2}{{\lef( {\int_\mathit \Omega {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}}- \lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} } \right)}^2}- \int_\mathit \Omega {F\lef( {x, u} \right)} } {\rm{d}}x \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{a}{2}\int_\mathit \Omega {\frac{{{{\left| {\Delta u} \right|}^{p\lef( x \right)}} - \lambda {{\left| u \right|}^{p\lef( x \right)}}}}{{p\lef( x \right)}}{\rm{d}}x} - \int_\mathit \Omega {F\lef( {x, u} \right)} {\rm{d}}x \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{a}{{2{p^ + }}}\int_\mathit \Omega {\lef( {{{\left| {\Delta u} \right|}^{p\lef( x \right)}} - \lambda {{\left| u \right|}^{p\lef( x \right)}}} \right){\rm{d}}x} - {C_7}\int_\mathit \Omega {{{\left| u \right|}^{\alpha \lef( x \right)}}} {\rm{d}}x - {C_8}{{\left\| u \right\|}_{{l^1}\lef( \Omega \right)}} \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{a}{{2{p^ + }}}{{\left\| u \right\|}^{{p^ - }}} - {C_7}\int_\mathit \Omega {{{\left| u \right|}^{\alpha \lef( x \right)}}} {\rm{d}}x - {C_9} \ge }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\frac{2}{{2{p^ + }}}{{\left\| u \right\|}^{{p^ - }}} - \lef( {{C_7} - {C_9}} \right), }&{{{\left| u \right|}_{\alpha \lef( x \right)}} \le 1}\\ {\frac{2}{{2{p^ + }}}{{\left\| u \right\|}^{{p^ - }}} - {C_7}{{\lef( {{\beta _k}\left\| u \right\|} \right)}^{{\alpha ^ + }}} - {C_9}, }&{{{\left| u \right|}_{\alpha \lef( x \right)}} > 1} \end{array} \ge } \right.}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left\| u \right\|}^{{p^ - }}}\left[{\frac{a}{{2{p^ + }}}-{C_7}{{\lef( {{\beta _k}} \right)}^{{\alpha ^ + }}}-{{\left\| u \right\|}^{{\alpha ^ + }-{p^ - }}}} \right] -{C_9}, } \end{array} $

取$ \left\| u \right\| = {r_k} = {\lef( {{C_7}\beta _k^{{\alpha ^ + }}\frac{{4{p^ + }}}{a}} \right)^{1/\lef( {{p^-}-{\alpha ^ + }} \right)}} $, 则有$ \varphi \lef( u \right) \ge r_k^{{p^-}}\frac{{4{p^ + }}}{a}-{C_9} $, 当k→∞时, βk→0, 且a>0, 有k→∞时, rk→+∞, 从而bk:=infu∈Zk, ‖u‖=rkφ(u)→+∞。

第3步验证泛函φ满足引理1.3(喷泉定理)中条件(ⅱ)。

当t≥t0>0时, 存在C10>0, 有$ at + \frac{b}{2}{t^2} \le {C_{10}}{t^2} $, 从而当t>0时, 有

$ at + \frac{b}{2}{t^2} \le {C_{10}}{t^2} + {C_{11}}。 $ (2.13)

由条件(W1), 存在δ > 0, |u| > δ, 充分大的C12 > 0, 有W(x, u)≥2C12|u|2p+。则

$ W\lef( {x,u} \right) \ge {C_{12}}{\left| u \right|^{2{p^ + }}} - {C_{13}}, $ (2.14)

对所有x∈Ω和u∈R成立, 其中C13=max{0, $ \mathop {\inf }\limits_{x \in \Omega, \left| u \right| \le \delta } $W(x, u)}。