上海理工大学学报  2022, Vol. 44 Issue (6): 583-587   PDF    
带有变号势函数和Hardy项的临界p-双调和方程弱解的存在性
崔会敏, 魏公明     
上海理工大学 理学院,上海 200093
摘要: 利用山路引理、集中紧性原理和Hardy不等式,研究了带有变号势函数和Hardy项的临界p-双调和方程弱解的存在性问题。首先验证了山路引理的几何条件,然后证明当 $0 < \mu < {\mu _0}$ ,山路水平 $c < \dfrac{2}{N} S^{N / 2 p}-\mu^{{p^*} /\left(p^*-q\right)}G$ 时满足 ${(PS)_c}$ 条件,最终证明了该类临界p-双调和方程至少存在一个非平凡弱解。
关键词: 山路引理     集中紧性原理     弱解     Hardy不等式    
The existence of weak solutions for critical p-biharmonic equations with sign-changing weight function and Hardy term
CUI Huimin, WEI Gongming     
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The existence of weak solutions for critical p-biharmonic equations with sign-changing weight function and Hardy term was studied by using mountain pass lemma, concentration-compactness principle and Hardy’s inequality. Firstly, the geometric conditions of mountain pass lemma are verified. Next, if $0 < \mu < {\mu _0}$ , ${(PS)_c}$ condition is satisfied when mountain level $c < \dfrac{2}{N} S^{N / 2 p}-\mu^{{p^*} /\left(p^*-q\right)}G$ . Thus, it is proved that there exists at least one nontrivial weak solution for the given critical ${{p - }}$ biharmonic equations.
Key words: mountain pass lemma     concentrative-compactness principle     weak solution     Hardy’s inequality    
1 问题的提出

四阶方程出现在数学和物理的各个分支中。例如,它可以用来研究悬索桥的行进波问题[1-2]和弹性板在流体中的静态扰度问题。近年来,非线性双调和方程和p-双调和方程解的存在性问题被广泛研究[3-4]。关于多重调和问题,Pucci等[5]研究了多重调和算子的临界指数和临界维数。Pucci等[6]研究了多重调和问题在 $N$ 维空间的一个球 $ {B_R} $ 上的非线性特征值问题,得到了特征值的连续谱,且证明了最小特征值是孤立的。特别地,对于带有Hardy项的奇异p-双调和椭圆型方程解的存在性问题正在被越来越多的学者关注[7-10]

2018年,文献[3]研究了带有临界指标和凹凸非线性项的p-双调和方程

$ \Delta _p^2u = \lambda f(x)|u{|^{q - 1}}u + |u{|^{{p^*} - 1}}u,\;\; x \in \varOmega $ (1)

其中, $u \in W_0^{2,p}(\varOmega )$ , $\varOmega \in {\mathbb{R}^N}$ 是一个有界光滑区域, $0 < q < 1 < p = \dfrac{{N + 4}}{{N - 4}},{\text{ }}N{\text{ > 4}}$ $\Delta _p^2$ 表示p-双调和算子。使用山路引理、集中紧性引理和亏格的方法证明了方程(1)分别在Dirichlet边界和Navier边界下解的多重性问题。类似地,在次临界下带有凹凸非线性函数和变号势函数的p-双调和问题在Navier边界下解的存在性也可以参考文献[11]。

2020年,文献[7]研究了带有Hardy项的p-双调和方程

$ \Delta _p^2u = \frac{{\mu |u{|^{r - 2}}u}}{{{{\left| x \right|}^s}}} + f(x,u),\;\;x \in \varOmega $ (2)

其中, $\varOmega \in {\mathbb{R}^N}$ 是一个有界光滑区域, $2 < 2p < N,$ ${\text{ }}\mu \geqslant 0,{\text{ }}0 \in \varOmega ,{\text{ }}$ ${{ p}} \leqslant {{r < }}{{{p}}^*}(s) = \dfrac{{(N - s)p}}{{N - 2p}} \leqslant {p^*}(0): = {p^*}$ 。使用山路引理等变分的方法,证明了方程(2)分别在Dirichlet边界和Navier边界下解的存在性和多重性。

受文献[3, 7, 11]的启发,本文主要研究了临界p-双调和方程

$ \Delta _p^2u - \lambda \frac{{|u{|^{p - 2}}u}}{{{{\left| x \right|}^{2p}}}} + \mu V(x)|u{|^{q - 2}}u = |u{|^{{p^ * } - 2}}u,{\text{ }}x \in {\mathbb{R}^N} $ (3)

其中, $u \in {W^{2,p}}({\mathbb{R}^N})$ $\Delta _p^2u = \Delta (|\Delta u{|^{p - 2}}\Delta u)$ 表示p-双调和算子, $2 \leqslant p < q < {p^ * }$ ${p^ * } = \dfrac{{Np}}{{N - 2p}}$ $N > 2p$ $V(x) \in {C_c}({\mathbb{R}^N})$ $V(x)$ 是变号函数。

本文所研究的问题比文献[7]的问题更一般,因为,在 ${\mathbb{R}^N}$ 上, ${W^{2,p}}({\mathbb{R}^N})$ ${L^{{p^*}}}({\mathbb{R}^N})$ 的嵌入不是紧的,恢复紧性是本文的重点也是难点。

2 预备知识

定义 $W$ 空间是 $C_0^\infty ({\mathbb{R}^N})$ 在范数 ${\left\| u \right\|_W} = {\left( { \displaystyle \int_{{\mathbb{R}^N}} {{{\left| {\Delta u} \right|}^p}{\rm{d}}} x} \right)^{\frac{1}{p}}}$ 下的完备化空间。定义 ${L^t}( {\mathbb{R}^N} )$ 上的范数: ${\left\| u \right\|_t} = {\left( \displaystyle \int_{{\mathbb{R}^N}} {{{\left| u \right|}^t}{\rm{d}}} x \right)^{\frac{1}{t}}}, {\text{ }}1 \leqslant t < \infty $

根据p-双调和Hardy’s不等式[12]

$ \int_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}} dx \leqslant \frac{1}{{\bar \lambda }}\int_{{\mathbb{R}^N}} {{{\left| {\Delta u} \right|}^p}} {\rm{d}}x,{\text{ }}\bar \lambda = {\left( {\frac{{\left( {n - 2p} \right)(p - 1)n}}{{{p^2}}}} \right)^p} $

可以推出,对任意的 $\lambda \in (0,\bar \lambda )$ $W$ 中的内积和范数等价于

$ (u,v) = {\int_{{\mathbb{R}^N}} {\left| {\Delta u} \right|} ^{p - 2}}\Delta u\Delta v{\rm{d}}x - \lambda \int_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^{p - 2}}uv}}{{{{\left| x \right|}^{2p}}}}} {\rm{d}}x $
$ \left\| u \right\| = {\left( {\int_{{\mathbb{R}^N}} \left({{\left| {\Delta u} \right|}^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x} \right)^{\frac{1}{p}}} $

$S$ 表示 $W$ 连续嵌入到 ${L^{{p^ * }}}({\mathbb{R}^N})$ 的最佳常数,即

$ S: = \inf\; \left\{ \int_{{\mathbb{R}^N}} \left({{\left| {\Delta u} \right|}^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x:{\text{ }}u \in W,{\text{ }}{\left\| u \right\|_{{p^ * }}} = 1\right\} $

根据文献[13]可知 $S$ 是可以取到的,假设存在正函数 $U \in W$ 可以取到 $S$ 且满足

$ \int_{{\mathbb{R}^N}} \left(|\Delta U{|^p} - \lambda \frac{{{{\left| U \right|}^p}}}{{{{\left| x \right|}^{2p}}}} \right){\rm{d}}x = \int_{{\mathbb{R}^N}} {|U{|^{{p^*}}}} {\rm{d}}x = {S^{\frac{N}{{2p}}}} $

定义函数 $V(x)$ 进一步满足 $\displaystyle \int_{{\mathbb{R}^N}} {V(x)|U{|^q}} {\rm{d}}x > 0$ ,不失一般性,令 $\varOmega = \left\{ {x:V(x) \ne 0} \right\}$

为了方便,当积分区域为 ${\mathbb{R}^N}$ 时,省略不写。令

$ \begin{split}& {M_1} = \frac{2}{N} > 0,{\text{ }}{{{M}}_2} = \left(\frac{1}{p} - \frac{1}{q}\right)C\left( {\mathop {\max }\limits_{x \in \bar \Omega }\; V(x)} \right) > 0\\& \qquad G = {M_2}{\left( {\frac{{q{M_2}}}{{{p^*}{M_1}}}} \right)^{\frac{q}{{{p^*} - q}}}} > 0 \end{split}$
$\begin{split}& {\mu _1} = {\left( {\frac{{2{S^{\frac{N}{{2p}}}}}}{{NG}}} \right)^{\frac{{{p^*} - q}}{{{p^*}}}}} > 0\\&{\text{ }}{\mu _2} = {\left( {\frac{{{t_1}^q\int_\Omega {V(x)|U{|^q}} {\rm{d}}x}}{{qG}}} \right)^{\frac{{{p^*} - q}}{{{p^*}}}}} > 0 \end{split} $

${\mu _0} = \min \;\{ {\mu _1},{\mu _2}\} > 0$ $C > 0,{{ }}{{{t}}_1} > 0$ ${t_1}$ 仅依赖于 ${\mu _1}$

本文研究问题(3)的弱解的存在性。对 $u \in W$ ,定义问题(3)的泛函:

$ I(u) = \frac{1}{p}\int_{} \left(|\Delta u| ^p - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x + \frac{\mu }{q}{\int_{} {V(x)|u|} ^q}{\rm{d}}x - \frac{1}{{{p^ * }}}{\int_{} {|u|} ^{{p^ * }}}{\rm{d}}x$

很容易得到 $I(u) \in {C^1}({\mathbb{R}^N},\mathbb{R})$ ,并且具有Fréchet导数为

$ \begin{split} \left\langle {I'(u),v} \right\rangle = &\int_{} \left({{\left| {\Delta u} \right|}^{p - 2}}\Delta u\Delta v - \lambda \frac{{{{\left| u \right|}^{p - 2}}uv}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x + \\&\mu {\int_{} {V(x)|u|} ^{q - 2}}uv{\rm{d}}x - {\int_{} {|u|} ^{{p^ * } - 2}}uv{\rm{d}}x \end{split} $

因为,问题(3)的所有解都是泛函 $I(u)$ 的临界点,也就是对任意的 $v \in W$ ,使得 $\left\langle {I'(u),v} \right\rangle = 0$ 成立的点,因此,可以通过求满足 $\left\langle {I'(u),v} \right\rangle = 0$ 的函数 $u$ ,就得到了问题(3)的弱解 $u$

本文主要结果为定理1。

定理1 如果 $0 < \mu < {\mu _0} = \min \{ {\mu _1},{\mu _2}\} ,{\text{ }}\lambda \in \left( {0,\bar \lambda } \right)$ ,问题(3)至少存在一个非平凡弱解。

3 定理1的证明

首先验证山路引理的几何条件,得到引理1。

引理1 对 $0 < \lambda < \bar \lambda $ ,有以下2个结论成立:

a. 存在常数 $\rho ,\;\beta > 0$ ,使得 $I(u) \geqslant \beta ,\left\| u \right\| = \rho $

b. 存在 $e \in W$ $\left\| e \right\| > \rho $ ,使得 $I(e) \leqslant 0$

证明  a. 因为, $V(x) \in {C_c}({\mathbb{R}^N})$ ,有

$ \begin{split} I(u) = & \frac{1}{p}\int_{} {\left(|\Delta u{|^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right)} {\rm{d}}x + \frac{\mu }{q}\int_{} {V(x)|u{|^q}} {\rm{d}}x - \frac{1}{{{p^ * }}}\int_{} {|u{|^{{p^ * }}}} {\rm{d}}x \geqslant \\& \frac{1}{p}{\left\| u \right\|^p} + \frac{\mu }{q}\left( {\mathop {\min }\limits_{x \in \bar \varOmega } \; V(x)} \right)\int_{} {|u{|^q}} {\rm{d}}x - \frac{1}{{{p^ * }}}{S^{ - \frac{{{p^ * }}}{2}}}{\left\| u \right\|^{{p^ * }}}\geqslant \\& \frac{1}{p}{\left\| u \right\|^p} + \frac{\mu }{q}\left( {\mathop {\min }\limits_{x \in \bar \varOmega } \;V(x)} \right){C^{ - \frac{q}{2}}}{\left\| u \right\|^q} - \frac{1}{{{p^ * }}}{S^{ - \frac{{{p^ * }}}{2}}}{\left\| u \right\|^{{p^ * }}} \end{split} $

因为, $ p < q < {p^ * } $ ,如果取 $ \left\| u \right\| = \rho > 0 $ 足够小,则一定存在 $ \;\beta > 0 $ ,使得 $ I(u) \geqslant \beta > 0 $

b. $ u \in W\backslash \{ 0\} $ ,则

$ \begin{split} I(tu) =& \frac{{{t^p}}}{p}\int_{} \left(|\Delta u{|^p} - \lambda \frac{{{{\left| u \right|}^p}}}{{{{\left| x \right|}^{2p}}}}\right){\rm{d}}x + \frac{{\mu {t^q}}}{q}\int_{} {V(x)|u{|^q}} {\rm{d}}x -\\& \frac{{{t^{{p^ * }}}}}{{{p^ * }}}\int_{} {|u{|^{{p^ * }}}}{\rm{d}}x \end{split}$

因为,当 $ t \to + \infty ,{\text{ }}I(tu) \to - \infty $ ,即一定可以取到 $ \left\| e \right\| > \rho $ ,使得 $ I(e) \leqslant 0 $ 。证毕。

通过引理1,结合山路引理[14],存在 $ {\left( {PS} \right)_c} $ 序列 $ \{ {u_n}\} \in W $ ,使得当 $ n \to \infty $ 时,

$ I({u_n}) \to c,{\text{ }}I'({u_n}) \to 0,\;\; c = {\inf _{h \in \Gamma }}{\sup _{t \in [0,1]}}I\left( {h\left( t \right)} \right) $
$ \varGamma = \left\{ {h \in C\left( {[0,1],W} \right):{\text{ }}h(0) = 0,{\text{ }}h(1) = e} \right\} $

定理1 当 $\mu \in (0,{\mu _1}),{\text{ }}\lambda \in \left( {0,\bar \lambda } \right)$ $c < \dfrac{2}{N}{S^{\frac{N}{{2p}}}} - G{\mu ^{^{\frac{{{p^*}}}{{{p^*} - q}}}}}$ 时, $I(u)$ 的任意 ${(PS)_c}$ 序列具有强收敛的子序列。

证明 假设 $\{ {u_n}\} \subset W$ 是泛函 $I(u)$ ${(PS)_c}$ 序列,则 $ I({u_n}) \to c,{\text{ }}I'({u_n}) \to 0 $ 成立,并且一定存在 $n \in \mathbb{N}, {\text{ }}M > 0$ ,使得

$ |I({u_n})| \leqslant M,{\text{ |}}I'({u_n}){u_n}| \leqslant M\left\| {{u_n}} \right\|$

$ \begin{split} M(1 + \left\| {{u_n}} \right\|) \geqslant & I({u_n}) - \frac{1}{q}I'({u_n}){u_n} = \left(\frac{1}{p} - \frac{1}{q}\right){\left\| {{u_n}} \right\|^p} + \\&\left(\frac{1}{q} - \frac{1}{{{p^ * }}}\right)\int_{} {|{u_n}{|^{{p^ * }}}} {\rm{d}}x \geqslant \left(\frac{1}{p} - \frac{1}{q}\right){\left\| {{u_n}} \right\|^p} \end{split}$

$\{ {u_n}\} \subset W$ 是一个有界列,此时可以假设存在一个 $u \in W$ ,使得在 $W$ ${u_n} \rightharpoonup u$ ,在 $L_{loc}^q({\mathbb{R}^N})(1 < q < {2^*})$ 中有 ${u_n} \to u$ ,利用集中紧性原理[14],有

$ \begin{split}& |\Delta {{{u}}_n}{{\text{|}}^p}{\text{dx}} \rightharpoonup {\text{d}}\gamma \geqslant {\left| {\Delta {{u}}} \right|^p}dx + \sum\limits_{k \in J} {{\gamma _k}{\delta _{{x_k}}}} + {\gamma _0}{\delta _0}\\& |{{{u}}_n}{{\text{|}}^{{p^ * }}}{\text{dx}} \rightharpoonup {\text{d}}\nu = {\left| u \right|^{{p^ * }}}dx + \sum\limits_{k \in J} {{\nu _k}{\delta _{{x_k}}}} + {\nu _0}{\delta _0} \\& \frac{{\left|{u}_{n}\right|}^{p}}{{\left|x\right|}^{2p}}\text{dx}\rightharpoonup d\sigma =\frac{{\left|u\right|}^{p}}{{\left|x\right|}^{2p}}\text{dx+}{\sigma }_{0}{\delta }_{0} \end{split} $ (4)

其中, $J \subset \mathbb{N}$ 是有限集, ${\delta _x}$ $x \in {\mathbb{R}^N}$ 时的Dirac测度, $ \gamma ,\text{  }\sigma \text{,  }\nu $ 是非负的有界测度。不失一般性,只考虑在奇点 $0 \in {\mathbb{R}^N}$ 处集中的可能性,对任意的 $\varepsilon > 0$ ,取截断函数 $\phi \in C_0^\infty ({\mathbb{R}^N})$ ,使得在 ${B_ \in }(0)$ 内有 $\phi {\text{ = }}1$ ,在 ${B_{2 \in }}(0)$ 外有 $\phi {\text{ = 0}}$ ,并且有 $\left|\nabla \varphi \right|\leqslant \dfrac{2}{\varepsilon },\text{  }\left|\Delta \phi \right|\leqslant \dfrac{2}{{\varepsilon }^{2}}$ 。因为,对 $W$ 中有界序列 $\left\{ {\phi {u_{\text{n}}}} \right\}$ ,必成立 $\mathop {\lim }\limits_{n \to \infty } \left\langle {I'({u_n}),\phi {u_n}} \right\rangle = 0$ ,于是,

$\begin{split}& \mathop {\lim }\limits_{n \to \infty } \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}} \Delta (\phi {u_n}){\rm{d}}x - \lambda \int_{} \phi {\rm{d}}\sigma {\text{ = }} \\&\qquad- \mu \int_{} {V(x)|u} {|^q}\phi {\rm{d}}x + \int_{} \phi {\rm{d}}\nu \end{split} $ (5)

因为,

$ \begin{split}& \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}\Delta (\phi {u_n})} {\rm{d}}x{\text{ = }}\int_{} {\Delta {u_n}{|^{p - 2}}\Delta {u_n}} \left( (\Delta \phi ){u_n} +\right.\\&\qquad \left.2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + \phi \Delta {u_n} \right){\rm{d}}x {{ = }}{\int_{} {\phi |\Delta {u_n}|} ^p}{\rm{d}}x + \\& \qquad \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta } {u_n}\left( {2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + {u_n}\Delta \phi } \right){\rm{d}}x{\text{ }} \\[-5pt] \end{split} $ (6)

对式(6)在 $n \to \infty $ 取极限,可得

$\begin{split}& \mathop {\lim }\limits_{n \to \infty } \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}\Delta (\phi {u_n})} {\rm{d}}x{\text{ = }}\int_{} {\phi {\rm{d}}\gamma } + \\&\qquad \mathop {\lim }\limits_{n \to \infty } \int_{} {|\Delta {u_n}{|^{p - 2}}\Delta } {u_n}\left( {2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + {u_n}\Delta \phi } \right){\rm{d}}x \\[-6pt] \end{split} $ (7)

现证 $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\;\lim }\limits_{n \to \infty } \;\int_{} {|\Delta {u_n}{|^{p - 2}}\Delta } {u_n}\left( {2\left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle + {u_n}\Delta \phi } \right)\cdot {\rm{d}}x = 0$

$\varepsilon \to 0$ 时,

$\begin{split}& 0 \leqslant \mathop {\lim }\limits_{n \to \infty } \;\left| {\int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}} \left\langle {\nabla \phi ,\nabla {u_n}} \right\rangle {\rm{d}}x} \right|\leqslant \\& \mathop {\lim }\limits_{n \to \infty } {\;\left({\int_{} {|\Delta {u_n}|} ^p}{\rm{d}}x\right)^{\frac{{p - 1}}{p}}}{\left(\int_{} {|\nabla \phi {|^p}|\nabla {u_n}{|^p}} {\rm{d}}x\right)^{\frac{1}{p}}} \leqslant \\& C\mathop {\lim }\limits_{n \to \infty } {\;\left(\int_{} {|\nabla \phi {|^p}|\nabla {u_n}{|^p}} {\rm{d}}x\right)^{\frac{1}{p}}} \leqslant \\& C{\left(\int_{\mathbb{R}^N} \cap B(0,2\varepsilon ) {|\nabla \phi {|^p}|\nabla u{|^p}} {\rm{d}}x\right)^{\frac{1}{p}}}\leqslant \\& {\text{ }} C{\left( \int_{\mathbb{R}^N} \cap B(0,2\varepsilon ) |\nabla \phi {|^N}{\rm{d}}x{)^{\frac{p}{N}}}\int_\mathbb{R}^N \cap B(0,2\varepsilon \right) {|\nabla u{|^{\frac{{Np}}{{N - p}}}}} {\rm{d}}x)^{\frac{{N - p}}{N}}} \leqslant \\& C\int_{B(0,2\varepsilon )} {|\nabla u} {|^{\frac{{Np}}{{N - p}}}}{\rm{d}}x{)^{\frac{{N - p}}{N}}} \to 0{\text{ }} \\[-15pt] \end{split} $ (8)

$ \begin{split} 0 \leqslant& \mathop {\lim }\limits_{n \to \infty } \;\left| {\int_{} {|\Delta {u_n}{|^{p - 2}}\Delta {u_n}} ({u_n}\Delta \phi ){\rm{d}}x} \right| \leqslant\\& {\lim _{n \to \infty }}{\int_{} {|\Delta {u_n}|} ^{p - 1}}|{u_n}||\Delta \phi |{\rm{d}}x \leqslant \\& \mathop {\lim }\limits_{n \to \infty } {\;\left(\int_{} {|\Delta {u_n}{|^p}} {\rm{d}}x\right)^{\frac{{p - 1}}{p}}}{\left(\int_{} {|{u_n}{|^p}|\Delta \phi {|^p}} {\rm{d}}x\right)^{\frac{1}{p}}}\leqslant \\& {{C}}{\left(\int_{{\mathbb{R}^N} \cap B(0,2\varepsilon )} {|\Delta \phi {|^{\frac{N}{2}}}} {\rm{d}}x\right)^{\frac{{2p}}{N}}}{\left(\int_{{\mathbb{R}^N} \cap B(0,2\varepsilon )} {|u{|^{{p^ * }}}} {\rm{d}}x\right)^{\frac{1}{{{p^ * }}}}} \leqslant \\& {\text{ }} {{C}}\left(\int_{B(0,2\varepsilon )} {|\Delta u{|^{{p^*}}}} {\rm{d}}x\right)^{\frac{1}{{{p^*}}}} \to 0\\[-12pt] \end{split} $ (9)

结合式(5)~(9),有

$ \begin{split} 0{\text{ = }}& \mathop {\lim }\limits_{\varepsilon \to 0} \left\{ - \int {\lambda \phi {\rm{d}}\sigma } + \int_{} {\phi {\rm{d}}\gamma } + \mu \int_{} {V(x)|u{|^q}{\rm{d}}x - \int_{} {\phi {\rm{d}}\nu } } \right\} \geqslant\\& - \lambda {\sigma _0} + {\gamma _0} - {\nu _0} \end{split} $

应用集中紧性原理[14],有 $ S{\left( {{\nu _0}} \right)^{\frac{p}{{{p^*}}}}} \leqslant {\gamma _0} - \lambda {\sigma _0} $ ,结合上式可以得到 ${\nu _0} \geqslant {S^{\frac{N}{{2p}}}}$

$V(x) \in {C_c}({\mathbb{R}^N})$ 可得, $ \displaystyle \int_{} {V(x)} |{u_n}{|^q}{\rm{d}}x \to \displaystyle \int_{} {V(x)} |u{|^q}{\rm{d}}x$ 。故有

$\begin{split} {c}=&\underset{n\to \infty }{\mathrm{lim}}\;I({u}_{n})=\underset{n\to \infty }{\mathrm{lim}}\;\left(I({u}_{n})-\frac{1}{p}\langle {I}^{\prime }({u}_{n}),{u}_{n}\rangle \right)=\\ &\underset{n\to \infty }{\mathrm{lim}}\left(\left(\frac{1}{q}-\frac{1}{p}\right)\mu {\displaystyle {\int }_{}V(x)|{u}_{n}{|}^{q}{\rm{d}}x+\left(\frac{1}{p}-\frac{1}{{p}^{\ast }}\right)}{{\displaystyle {\int }_{}\left|{u}_{n}\right|}}^{{p}^{\ast }}{\rm{d}}x\right)=\\ &\left(\frac{1}{q}-\frac{1}{p}\right)\mu {\displaystyle {\int }_{}V(x)|{u}_{n}{|}^{q}{\rm{d}}x+\underset{n\to \infty }{\mathrm{lim}}\left(\frac{1}{p}-\frac{1}{{p}^{\ast }}\right)}{{\displaystyle {\int }_{}\left|{u}_{n}\right|}}^{{p}^{\ast }}{\rm{d}}x\geqslant\\& \left(\frac{1}{q}-\frac{1}{p}\right)\mu {\displaystyle {\int }_{}V(x)|u{|}^{q}{\rm{d}}x+\left(\frac{1}{p}-\frac{1}{{p}^{\ast }}\right)}\left({\displaystyle {\int }_{}|u{|}^{{p}^{\ast }}{\rm{d}}x+{\nu }_{0}}\right)\geqslant\\ & \left(\frac{1}{q}-\frac{1}{p}\right)\mu C\left(\underset{x\in \overline{\varOmega }}{\mathrm{max}}V(x)\right)\left(\displaystyle {\int }_{}|u{|}^{{p}^{\ast }}{\rm{d}}x\right)^{\frac{q}{{p}^{\ast }}}+\\ &\frac{2}{N}{\displaystyle {\int }_{}|u{|}^{{p}^{\ast }}{\rm{d}}x+\frac{2}{N}{S}^{\frac{N}{2p}}} \end{split} $

其中, $C$ 为正常数。令 $g(x) = :{M_1}{x^{{p^ * }}} - {M_2}\mu {x^q}$

$ {M}_{1}\text=\frac{2}{N}\text{,}\text{ }{M}_{2}=\left(\frac{1}{p}-\frac{1}{q}\right)C\left(\underset{x\in \overline{\varOmega }}{\mathrm{max}}\;V(x)\right) $

$g'(x) = 0$ 很容易求得 $g(x)$ 的临界点为 ${x_0} = {\left( {\dfrac{{q{M_2}\mu }}{{{p^ * }{M_1}}}} \right)^{\frac{1}{{{p^ * } - q}}}}$ 。通过计算可知 $g(x)$ ${x_0} = {\left( {\dfrac{{q{M_2}\mu }}{{{p^ * }{M_1}}}} \right)^{\frac{1}{{{p^ * } - q}}}}$ 上取得极小值 $g({x_0}){\text{ = }}{M_3} - \mu {M_4}$ ,其中,

$ {M_3}{\text{ = }}{M_1}{\left(\frac{{q{M_2}\mu }}{{{p^*}{M_1}}}\right)^{\frac{{{p^*}}}{{{p^*} - q}}}}\text{,}{M_4}{\text{ = }}{M_2}{\left(\frac{{q{M_2}\mu }}{{{p^*}{M_1}}}\right)^{\frac{q}{{{p^*} - q}}}} $

因此,可得

$\begin{split} & c \geqslant \frac{2}{N}{S^{\frac{N}{{2p}}}}{\text{ + }}{M_3} - \mu {M_4} = \frac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G\\&G = {M_2}{\left(\frac{{q{M_2}}}{{{p^*}{M_1}}}\right)^{\frac{q}{{{p^*} - q}}}} \end{split}$

与条件矛盾。故当 $n \to \infty $ 时,有 $ \displaystyle \int_{} {{\text{|}}{u_n}{|^{{p^*}}}} {\rm{d}}x \to \displaystyle \int_{} {{\text{|}}u{|^{{p^*}}}} {\rm{d}}x$ 。根据文献[15]的引理3.4,存在常数 $C > 0$ ,使得

$ \begin{split} o(1) = &\left\langle {I'({u_n}) - I'(u),{u_n} - u} \right\rangle {\text{ = }}\\ & \int_{} {\left( {|\Delta {u_n}{|^{p - 2}}{u_n} - |\Delta u{|^{p - 2}}u} \right)} \left( {{u_n} - u} \right){\rm{d}}x +\\ & \mu \int_{} {V(x)} (|{u_n}{|^{q - 2}}{u_n} - |u{|^{q - 2}}u)({u_n} - u){\rm{d}}x - \\ & \int_{} ( |{u_n}{|^{{p^*} - 2}}{u_n} - |u{|^{{p^*} - 2}}u)({u_n} - u){\rm{d}}x\geqslant \\ & {{C}}{\left\| {{u_n} - u} \right\|^p} + o(1) \end{split}$

故在 $W$ 中有 ${u_n} \to u$ 。证毕。

定理2 当 $\mu \in (0,{\mu _0}),{\text{ }}\lambda \in {\text{(0,}}\bar \lambda {\text{)}}$ ,山路水平 $c$ 满足 $0 < \beta \leqslant c < \dfrac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$

证明 对任意的 $\mu \in (0,{\mu _1}),{\text{ }}\lambda \in {\text{(0,}}\bar \lambda {\text{)}}$ $\dfrac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*} - q}}{{{p^*}}}}}G > 0$

定义函数

$ h(t) = \frac{1}{p}{t^p}{\left\| U \right\|^p} - \frac{{{t^{{p^*}}}}}{{{p^*}}}\int_{} {|U{|^{{p^*}}}} {\rm{d}}x = :{C_1}{t^p} - {C_2}{t^{{p^*}}},\;\;t \geqslant 0$

其中, ${C_1} = \dfrac{1}{p}{S^{\frac{N}{{2p}}}},{\text{ }}{{{C}}_2} = \dfrac{1}{{{p^*}}}{S^{\frac{N}{{2p}}}}$ ,通过计算很容易得到 $h(t)$ 存在唯一的零点 ${t_0} = {\left(\dfrac{{p{C_1}}}{{{p^*}{C_2}}}\right)^{\frac{1}{{{p^*} - p}}}} = 1$ ,并且 $h(t)$ ${t_0}$ 处取得极大值 $h(t) = \dfrac{2}{N}{S^{\frac{N}{{2p}}}}$ ,即对任意的 $t > 0$ ,都有

$ \begin{split} I(tu) =& h(t) - \frac{{\mu {t^q}}}{q}\int_{} V(x)|U{|^q}{\rm{d}}x \leqslant \\& \frac{2}{N} {S^{\frac{N}{{2p}}}} - \frac{{\mu {t^q}}}{q}\int_{} {V(x} )|U{|^q}{\rm{d}}x \end{split}$ (10)

因为, $I(0) = 0$ ,则存在仅依赖于 ${\mu _1}$ ${t_1} > 0$ ,使得对任意的 $\mu \in (0,{\mu _1})$ ,都有

$ \mathop {\max }\limits_{0 \leqslant t \leqslant {t_1}}\; I(tu) < \frac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$

另一方面,由式(10)可知,

$ \mathop {\max }\limits_{t > {t_1}} \;I(tu) \leqslant \frac{2}{N}{S^{\frac{N}{{2p}}}} - \frac{{\mu {t_1}^q}}{q}\int_{} {V(x} )|U{|^q}{\rm{d}}x $

即对任意的 $\mu \in (0,{\mu _2})$ ,都有 $\mathop {\max }\limits_{t > {t_1}} \;I(tu) < \dfrac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$

综上,取 ${\mu _0} = \min \{ {\mu _1},\;{\mu _2}\}$ ,对任意的 $\mu \in (0,\;{\mu _0})$ ,都有 $\mathop {\max }\limits_{t > 0} \;I(tu) < \dfrac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$

根据山路水平 $c$ 的定义可知, $c \leqslant \mathop {\max }\limits_{t > 0} \;I(tu) < \dfrac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$ 。证毕。

通过以上定理和引理,可知当 $\mu \in (0,{\mu _0}),{\text{ }}{\mu _0} = \min\; \{ {\mu _1},{\mu _2}\} ,{\text{ }}\lambda \in {\text{(0,}}\bar \lambda {\text{)}}$ ,山路水平 $c < \dfrac{2}{N}{S^{\frac{N}{{2p}}}} - {\mu ^{\frac{{{p^*}}}{{{p^*} - q}}}}G$ ,泛函 $I(u)$ 的任意一个 ${(PS)_c}$ 序列都有一个强收敛的子序列,即方程(3)在 $W$ 中的弱解是存在的,并且弱解最少有一个。

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