源于Capillarity方程与毛细现象等实际问题密切相关, 所以对这类问题的研究活跃在数学领域.针对Capillarity方程解的存在性及其特征值问题的研究, 常见的方法是山路原理和极大极小原理.近期, 在文[1-2]中, 作者提出了新方法-利用极大单调算子和$m$增生映射的值域理论展开了对这类问题的研究.具体地, 2013年, 作者在文[1]中给出了以下具Neumann边值条件的Capillarity方程在$W^{1, p}(\Omega)$空间中存在解的充分条件

$ \begin{eqnarray}\left\{ \begin{array}{lll} -{\rm div}[(1+\frac{|\nabla u|^p}{\sqrt{1+|\nabla u|^{2p}}})|\nabla u|^{p-2}\nabla u]+ \lambda (|u|^{q-2}u + |u|^{r-2}u) = f(x)~~{\rm a.e.}~~x\in \Omega, \\ - <\vartheta, (1+\frac{|\nabla u|^p}{\sqrt{1+|\nabla u|^{2p}}})|\nabla u|^{p-2}\nabla u> \in \beta(u(x))-h(x)~~{\rm a.e.}~~x\in \Gamma, \end{array} \right.\end{eqnarray} $

(1)

其主要研究思路是首先建立具Neumann边值条件的Capillarity方程和具Dirichlet边值条件的Capillarity方程之间的关系; 然后将边值条件转化为非线性极大单调算子之和的形式; 最后利用Reich ^[3]提出的极大单调算子值域几乎相等的结论进行讨论.

2012年, 作者在文[2]中利用Calvert-Gupta ^[4]提出的$m$增生映射值域的扰动结论研究了以下具Neumann边值条件的Capillarity方程在$L^{p}(\Omega)$空间中存在解的结论

$ \begin{equation}\left\{ \begin{array}{lll} -{\rm div}[(1+\frac{|\nabla u|^p}{\sqrt{1+|\nabla u|^{2p}}})|\nabla u|^{p-2}\nabla u]+ \lambda (|u|^{q-2}u + |u|^{r-2}u)+\\g(x, u(x))= f(x) ~~{\rm a.e.}~~x\in \Omega, \\ - <\vartheta, (1+\frac{|\nabla u|^p}{\sqrt{1+|\nabla u|^{2p}}})|\nabla u|^{p-2}\nabla u> \in \beta_x(u(x))~~{\rm a.e.}~~x\in \Gamma. \end{array} \right.\end{equation} $

(1.2)

其主要研究思路是将边值条件和非线性方程揉合在一起定义所需的$m$增生映射.

当某个实际问题需要用多个Capillarity方程来研究的时候, 就引出了研究Capillarity系统的问题.本文分为两部分.第一部分是引言和预备知识, 第二部分给出一类Capillarity系统存在非平凡解的充分条件.本质上将对Capillarity方程的研究推广到了方程组的情形, 并采用了不同于文[1, 2]的方法.

下面介绍预备知识.

设$E$为实Banach空间, $E^*$为其对偶空间.正规对偶映射$J: E \rightarrow 2^{E^*}$定义为$ J(x)=\{x^*\in E^* : \langle x, x^*\rangle=\|x\|^2=\|x^*\|^2\}, \forall x \in E, $其中$\langle \cdot, \cdot\rangle$表示$E$与$E^*$元素间的广义对偶对.众所周知, 当$E$为实严格凸Banach空间时, 正规对偶映射为单值映射.为方便起见, 仍用J表示单值正规对偶映射.

称多值算子$B: E \rightarrow 2^{E^*}$为单调算子^[5]:若$ \forall x_{i} \in D(B), y_{i}\in Bx_{i}, i =1, 2, $均有

$ \langle x_{1}-x_{2}, y_{1}-y_{2}\rangle \geq 0. $

(1.3)

若(1.3) 式中等号成立的充要条件是$x_1 = x_2, $则称$B$为严格单调算子.称算子$B$为极大单调算子:若$B$单调且$\forall r > 0, R(J+rB) = E^*$.

称多值映射$A:E \rightarrow 2^E$为增生映射^[6]:若$\langle v_{1}-v_{2}, j(u_{1}-u_{2})\rangle\ge 0, $ $\forall u_{i} \in D(A)$和$v_{i} \in Au_{i}, \, i=1, 2$, 这里$j(u-v) \in J(u-v).$称增生映射$A$为$m$增生的:若$R(I+\lambda A) = E, $ $\forall \lambda>0.$

设$E_1$和$E_2$均为实Banach空间.映射$C : E_1 \rightarrow E_2$称为有界映射:若$C$将$E_1$中的有界子集映射成$E_2$中的有界子集.映射$C : E_1 \rightarrow E_2$称为紧映射:若$C$连续且将$E_1$中的有界子集映射成$E_2$中的相对紧集.

称函数$\Phi : E \rightarrow (-\infty, +\infty]$为正则凸函数^[6]:若存在$u_0 \in E$使得$\Phi(u_0) <+\infty$且$\Phi((1-\lambda)u + \lambda v)\leq (1-\lambda)\Phi(u)+\lambda \Phi(v), $ $\forall u, v \in E$及$\lambda \in [0, 1].$称函数$\Phi : E \rightarrow (-\infty, +\infty]$是下半连续函数:若$\lim\limits_{y \rightarrow x}\inf\Phi(y) \geq \Phi(x), $$\forall x\in E.$对$E$上定义的正则凸函数$\Phi$, 其次微分$\partial \Phi : E \rightarrow E^* $定义为$\forall u \in E, $

$ \partial \Phi(u) = \{w \in E^*: \Phi(u)-\Phi(v)\leq \langle u -v, w\rangle, ~~\forall v \in E\}. $

设$E_1$和$E_2$为实Banach空间.则记号“$E_1 \hookrightarrow \hookrightarrow E_2$”表示空间$E_1$紧嵌入到$E_2$.

定义1.1 ^[5] 设$N$为正整数, $\Omega$为$R^N$中有界开集, 称$g:\Omega \times R^N \rightarrow R $满足Caratheodory条件, 如果

(ⅰ) $g(x, \cdot ): R^{N} \rightarrow R$连续a.e. $x \in \Omega$;

(ⅱ) 映射$g(\cdot, r ): \Omega \rightarrow R$可测, $\forall r \in R^N$.

引理1.1^[7] 设$X_1$, $X_2, \cdots, X_M$为实Banach空间, 则$X_1 \times X_2\times \cdots \times X_M$定义为

$ X_1 \times X_2\times \cdots \times X_M = \{(x_1, x_2, \cdots, x_M): x_i \in X_i, ~ i = 1, 2, \cdots, M\}. $

$\forall (x_1, x_2, \cdots, x_M), (y_1, y_2, \cdots, y_M) \in X_1 \times X_2\times \cdots\times X_M, $定义

$ k_1(x_1, x_2, \cdots, x_M)+k_2(y_1, y_2, \cdots, y_M)= (k_1x_1+k_2y_1, k_1x_2+k_2y_2, \cdots, k_1x_M+k_2y_M), $

则$X_1 \times X_2\times \cdots \times X_M$成为线性空间.特别地, 当定义如下范数

$ \|(x_1, x_2, \cdots, x_M)\|= (\|x_1\|^2_{X_1}+\|x_2\|^2_{X_2}+\cdots+\|x_M\|^2_{X_M})^{\frac{1}{2}} $

时, $X_1 \times X_2 \times \cdots\times X_M$为实Banach空间且$(X_1 \times X_2 \times \cdots\times X_M)^{*} = X_1^{*} \times X_2^{*}\times \cdots \times X_M^{*}.$

引理1.2^[5] 若$\Phi : E \rightarrow (-\infty, \infty]$为正则凸、下半连续函数, 则$\partial \Phi : E \rightarrow 2^{E^*}$极大单调.

引理1.3^[4] 令$\Omega$为$R^{N}$中的有界区域, 令$J_{p}:L^{p}(\Omega)\rightarrow L^{p'}(\Omega)$表示正规对偶映射.则当$ 1 < p <+\infty$时,

$ J_{p}u = |u|^{p-1}{\rm sgn} u\|u\|^{2-p}_{p}, ~~\forall u\in L^{p}(\Omega), $

这里$\frac{1}{p}+\frac{1}{p'}=1.$

引理1.4^[8] 令$ \Omega$为$R^{N}$中的有界锥形区域.若$mp>N, $则$W^{m, p}(\Omega)\hookrightarrow \hookrightarrow C_{B}(\Omega);$若$ 0 < mp\leq N $且$ q_{0}= \frac{Np}{N-mp}, $则$ W^{m, p}(\Omega)\hookrightarrow\hookrightarrow L^{q}(\Omega), $其中$1 \leq q < q_{0}.$

定理1.1^[9] 令$E$为实Banach空间, $A: E \rightarrow 2^E$为$m$增生映射且$(I+A)^{-1}: E \rightarrow E$为紧映射.令$C: D(A) \subset E \rightarrow E$为有界映射且存在$\lambda \in (0, 1]$使得$C(I+\lambda A)^{-1}: E \rightarrow E$为紧映射.假设$p \in E$且存在正常数$b, r$和$z \in D(A) : \|z\| < b$满足

$ \langle Cx + y - p, j(x-z)\rangle \geq 0, $

$\forall x \in D(A): \|x\|\geq r, $ $\forall y \in Ax, $其中$j(x-z) \in J(x-z)$, 那么$ p \in R(A+C).$

本文研究以下Capillarity系统

$ \begin{equation}\left\{ \begin{array}{lll} -{\rm div}[(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2{p_i}}}})|\nabla u_i|^{p_i-2}\nabla u_i]+ \lambda_i (|u_i|^{q_i-2}u_i + |u_i|^{r_i-2}u_i) +\\ \varepsilon_i g_i(x, \nabla u_i, u_i)= f_i(x)\\ ~~{\rm a.e.}~~x\in \Omega, \\ - <\vartheta, (1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2{p_i}}}}) |\nabla u_i|^{p_i-2}\nabla u_i> \in \beta_x(u_i(x))~~{\rm a.e.}~~x\in \Gamma, \\ i = 1, 2, \cdots, M. \end{array} \right.\end{equation} $

(2.1)

在(2.1) 式中, $\Omega$是$R^{N}$$ (N \geq 1)$中有界锥形区域且其边界$\Gamma\in C^{1}$ (见文献[10]); $\varepsilon_i \in R^+\bigcup \{0\}, $ $\lambda_i \in R^+, $ $i = 1, 2, \cdots, M;$ $\vartheta$表示$\Gamma$的外法向导数; $|\cdot|$表示$R^{N}$中的范数; $\langle\cdot, \cdot \rangle$表示$R^N$中的内积.

$g_i:\Omega\times R^N \times R \rightarrow R$为满足Carathéodory条件的给定函数, $i = 1, 2, \cdots, M.$假设它们满足以下增长性条件:存在正常数$b_i$使得

$ |g_i(x, r_1, \cdots, r_{N+1})-g_i(x, s_1, \cdots, s_{N+1})|\leq b_i|r_{N+1}-s_{N+1}|, $

$ \forall(r_1, r_2, \cdots, r_{N+1}), (s_1, s_2, \cdots, s_{N+1})\in R^{N+1}, $ $i = 1, 2, \cdots, M.$

$\beta_{x}$为$\varphi_{x}$的次微分, 即$\beta_x\equiv\partial\varphi_x.$这里$\varphi_{x}= \varphi(x, \cdot):R\rightarrow R, $ $\forall x\in\Gamma, $其中$\varphi:\Gamma\times R\rightarrow R$为给定函数, 见文[11].假设$\forall x\in\Gamma, ~\varphi_{x}: R \rightarrow R$为正则凸、下半连续函数, $\varphi_{x}(0)=0$, $0\in \beta_x(0)$且$\forall t\in R, $函数$x\in\Gamma \rightarrow (I+\lambda \beta_x)^{-1}(t)\in R$可测, $\lambda >0.$

假设对$i = 1, 2, \cdots, M, $ $ \frac{2N}{N+1} < p_i < +\infty;$当$p_i \geq N$时, $1 \leq q_i, r_i < +\infty;$当$p_i < N$时, $1 \leq q_i, r_i \leq \frac{Np_i}{N-p_i}.$进一步假设Green公式成立.

引进如下记号: $Y = L^{p_1}(\Omega)\times L^{p_2}(\Omega)\times \cdots \times L^{p_M}(\Omega).$分别用$\|\cdot\|_{p_i}$和$\|\cdot\|_Y$表示$L^{p_i}(\Omega)$和空间$Y$中的范数.令$ \frac{1}{p_i} + \frac{1}{p'_i} = 1, $ $i = 1, 2, \cdots, M.$

引理2.1  定义$\Phi_{i}:W^{1, p_i}(\Omega)\rightarrow R$为$\forall u\in W^{1, p_i}(\Omega)$,

$ \Phi_{i}(u) = \int_{\Gamma}\varphi_{x}(u|_{\Gamma}(x))d\Gamma(x), $

则

(ⅰ) $\Phi_{i}$是$W^{1, p_i}(\Omega)$上的正则凸、下半连续函数.从而引理1.2蕴含$\partial \Phi_i$是极大单调算子, $i = 1, 2, \cdots, M.$

(ⅱ) $\langle v_i, \partial\Phi_i(u_i)\rangle = \displaystyle\int_\Omega \beta_x(u_i|_\Gamma)v_i|_\Gamma d\Gamma(x), $ $\forall u_i, v_i \in W^{1, p_i}(\Omega), $ $i = 1, 2, \cdots, M.$因此由$0 \in \beta_x(0)$可知$\partial\Phi_i(0)= 0, ~i = 1, 2, \cdots, M.$

(ⅲ) $\langle \psi, \partial\Phi_i(u_i)\rangle = 0, $ $\forall \psi \in C_0^\infty(\Omega).$

证 由类似于文[10]可证结论(ⅰ)和(ⅲ)成立.类似于文[12]可证结论(ⅱ)成立.

引理2.2^[2] 定义$B_{i}:W^{1, p_i}(\Omega)\rightarrow (W^{1, p_i}(\Omega))^*$为

$ \langle v, B_{i}u \rangle = \int_{\Omega}\langle(1+\frac{|\nabla u|^{p_i}}{\sqrt{1+|\nabla u|^{2p_i}}})|\nabla u|^{p_i-2}\nabla u, \nabla v \rangle dx\\ + \lambda_i \int_{\Omega}|u(x)|^{q_i-2}u(x) v(x) dx + \lambda_i \int_{\Omega}|u(x)|^{r_i-2}u(x) v(x)dx, $

$ \forall u, v \in W^{1, p_i}(\Omega).$则$B_{i}$极大单调且严格单调, $i = 1, 2, \cdots, M.$

引理2.3^[2] 定义$A_{i}:L^{p_i}(\Omega)\rightarrow L^{p_i}(\Omega)$为$ D(A_{i}) = \{u\in L^{p_i}(\Omega)|$存在$ f \in L^{p_i}(\Omega), $使得$f = B_{i}u+\partial\Phi_{i}(u) \}.$对$ u \in D(A_{i}), $令$ A_{i}u = \{ f \in L^{p_i}(\Omega)|f = B_{i}u + \partial\Phi_{i}(u)\}$, $i = 1, 2, \cdots, M$.则$A_i$是$m$增生映射, $i = 1, 2, \cdots, M$.

命题2.1 定义$ A: Y \rightarrow Y$为

$ Au = (A_1u_1, A_2u_2, \cdots, A_Mu_M), ~~u = (u_1, u_2, \cdots, u_M)\in Y, $

则$A$是$m$增生映射.

证 首先注意当$\frac{2N}{N+1}< p_i < +\infty$时, $L^{p_i}(\Omega)$为严格凸Banach空间, 所以定义在$L^{p_i}(\Omega)$上的正规对偶映射为单值映射, $i =1, 2, \cdots, M$.

若定义$J: Y \rightarrow Y^*$为

$ J u = (J_{p_1}u_1, J_{p_2}u_2, \cdots, J_{p_M}u_M), $

$\forall u = (u_1, u_2, \cdots, u_M)\in Y, $其中$J_{p_i}$表示$L^{p_i}(\Omega)$上的正规对偶映射, $i = 1, 2, \cdots, M, $则$J$为$Y$上的正规对偶映射.

事实上, 因为$J_{p_i}$表示$L^{p_i}(\Omega)$上的正规对偶映射, 所以由乘积空间的性质和正规对偶映射的定义可知

$ \langle u , Ju\rangle = \langle u_1, J_{p_1}u_1 \rangle +\langle u_2, J_{p_2}u_2\rangle +\cdots +\langle u_M, J_{p_M}u_M\rangle\\ = \|u_1\|^2_{p_1}+\|u_2\|^2_{p_2}+\cdots+\|u_M\|^2_{p_M} = \|u\|_{Y}^2, $

而且

$ \|Ju\|^2_{Y^*} = \|J_{p_1}u_1\|^2_{p'_1}+\|J_{p_2}u_2\|^2_{p'_2}+\cdots +\|J_{p_M}u_M\|^2_{p'_M}=\\ \|u_1\|^2_{p_1}+\|u_2\|^2_{p_2}+\cdots +\|u_M\|^2_{p_M}= \|u\|^2_Y. $

因此$J$为$Y$上的正规对偶映射.

因为$A_i$是增生映射, 所以$\forall u = (u_1, u_2, \cdots, u_M), v = (v_1, v_2, \cdots, v_M) \in Y, $有

$ \langle Au - Av, J(u-v) \rangle =\langle A_1u_1 - A_1v_1, J{p_1}(u_1-v_1) \rangle + \langle A_2u_2 - A_2v_2, J{p_2}(u_2-v_2)\rangle \\ \;\;\;\;\;\;\;\;\;+\cdots+ \langle A_Mu_M - A_Mv_M, J{p_M}(u_M-v_M) \rangle\geq 0. $

因此$A$为增生映射.

$\forall~ \lambda >0$及$\forall v = (v_1, v_2, \cdots, v_M) \in Y, $因$A_i$是$m$增生映射, 故存在$u_i \in L^{p_i}(\Omega)$使得$v_i = u_i + \lambda A_iu_i, $于是$R(I+\lambda A) = Y.$

至此证明了$A$是$m$增生映射.

引理2.4 若$\eta_{\mu}^{(i)}: R \rightarrow R$单调、Lipschitz连续具Lipschitz常数$\frac{1}{\mu}$且$(\eta_{\mu}^{(i)})'$在$R$上除至多有限点外连续, 其中$\mu > 0, $则$\langle \eta_{\mu}^{(i)}(u - v), \partial\Phi_{i}(u)- \partial\Phi_{i}(v)\rangle \geq 0, ~\forall u \in W^{1, p_i}(\Omega), ~ i = 1, 2, \cdots, M.$进一步由引理2.1知$\langle \eta_{\mu}^{(i)}(u), \partial\Phi_{i}(u)\rangle \geq 0, ~\forall u \in W^{1, p_i}(\Omega), ~ i = 1, 2, \cdots, M.$

证 因为$\eta_{\mu}^{(i)}: R \rightarrow R$单调且Lipschitz常数为$\frac{1}{\mu}$, 所以存在$\kappa_x^{(i)} \in [0, 1]$满足

$ u|_\Gamma (x) - \mu\eta_{\mu}^{(i)}(u|_\Gamma (x)-v|_\Gamma (x))= \kappa_x^{(i)} u|_\Gamma (x)+(1-\kappa_x^{(i)})v|_\Gamma (x) $

和

$ v|_\Gamma (x) + \mu\eta_{\mu}^{(i)}(u|_\Gamma (x)-v|_\Gamma (x))= \kappa_x^{(i)} v|_\Gamma (x)+(1-\kappa_x^{(i)})u|_\Gamma (x). $

由假设条件知$\varphi_x$为凸函数, 从而

$ \varphi_x(u|_\Gamma (x) - \mu\eta_{\mu}^{(i)}(u|_\Gamma (x)-v|_\Gamma (x)))+\varphi_x(v|_\Gamma (x) +\\ \mu\eta_{\mu}^{(i)}(u|_\Gamma (x)-v|_\Gamma (x))) \leq \varphi_x(u|_\Gamma (x))+\varphi_x(v|_\Gamma (x)). $

利用引理2.1知

$ \Phi_i (u - \mu \eta^{(i)}_{\mu}(u - v))+ \Phi_i (v + \mu \eta^{(i)}_{\mu}(u - v))\leq \Phi_i(u)+\Phi_i(v). $

由次微分的定义可知

$ \Phi_i(u - \mu \eta^{(i)}_{\mu}(u - v))-\Phi_i(u) \geq \langle - \mu \eta^{(i)}_{\mu}(u - v), \partial\Phi_i(u)\rangle $

和

$ \Phi_i(v+ \mu \eta^{(i)}_{\mu}(u - v))-\Phi_i(v) \geq \langle \mu \eta^{(i)}_{\mu}(u - v), \partial\Phi_i(v)\rangle. $

从而

$ \mu \langle\eta^{(i)}_{\mu}(u - v), \partial\Phi_i(v)- \partial\Phi_i(u) \rangle \leq \Phi_i(u - \mu \eta^{(i)}_{\mu}(u - v))+\\\Phi_i(v+ \mu \eta^{(i)}_{\mu}(u - v))-\Phi_i(u)-\Phi_i(v) \leq 0. $

因此结论成立.

题2.2 映射$(I+A)^{-1}: Y \rightarrow Y$为紧映射.

证 若$u + Au = w $且$\{w = (w_1, w_2, \cdots, w_M)\}$在$Y$中有界, 则只需证明$\{u = (u_1, u_2, \cdots, u_M)\}$在$Y$中相对紧.

事实上, 转化为证明若$u_i + A_iu_i = w_i $且$\{w_i\}$在$L^{p_i}(\Omega)$中有界, 则$\{u_i\}$在$L^{p_i}(\Omega)$中相对紧, $i = 1, 2, \cdots, M.$

将证明分为以下两种情形:

(ⅰ) $p_i \geq 2.$

对$ k > 0, $定义$\delta_{k}^{(i)}: R \rightarrow R$为$\delta_{k}^{(i)}(t) = |(t\bigwedge k)\bigvee (-k)|^{p_i-1}{\rm sgn}t.$则$\delta_{k}^{(i)}$单调、Lipschitz连续、$\delta_{k}^{(i)}(0) = 0$且$(\delta_{k}^{(i)})'$在$R$上除至多有限点外连续.引理2.4蕴含

$ \langle \delta_{k}^{(i)}u_i, A_iu_i \rangle = \langle\delta_{k}^{(i)}(u_i), B_i u _i\rangle + \langle \delta_{k}^{(i)}(u_i), \partial\Phi_i(u_i)\rangle\\ \;\;\;\;\;\geq \langle\delta_{k}^{(i)}(u_i), B_i u _i\rangle \geq \int_\Omega |\nabla u_i|^{p_i}[(\delta_{k}^{(i)})'(u_i)]dx \\ \;\;\;\;\;\;\geq {\rm const} \int_{\{|u_i| < k\}}|{\rm grad}(|(u_i\bigwedge k)\bigvee (-k)|^{2-\frac{2}{p_i}}{\rm sgn}u_i)|^{p_i}dx. $

因此

$ \langle |u_i|^{p_i-1}{\rm sgn}u_i, A_iu_i \rangle = \lim_{k \rightarrow \infty} \langle \delta_{k}^{(i)}u_i, A_iu_i \rangle \nonumber\\ \;\;\;\;\;\;\;\geq {\rm const}~\int_{\Omega}|{\rm grad}(|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i)|^{p_i}dx. $

(2.2)

(2.2) 式蕴含

$ \|w_i\|_{p_i}\|u_i\|^{\frac{p_i}{p_i'}}_{p_i}\geq \langle |u_i|^{p_i-1}{\rm sgn}u_i, w_i \rangle\nonumber\\ \;\;\;\;\;\;\;= \langle |u_i|^{p_i-1}{\rm sgn}u_i, u_i \rangle +\langle |u_i|^{p_i-1}{\rm sgn}u_i, A_iu_i \rangle \nonumber\\ \;\;\;\;\;\;\;\geq \|u_i\|_{p_i}^{p_i}+{\rm const}~\int_{\Omega}|{\rm grad}(|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i)|^{p_i}dx. $

(2.3)

由(2.3) 式可知

$ {\rm const}.\geq \|w_i\|_{p_i}\geq \|u_i\|_{p_i}^{p_i-\frac{p_i}{p_i'}}= \|u_i\|_{p_i} = \||u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i\|^{\frac{p_i}{2(p_i-1)}}_{\frac{p_i^2}{2(p_i-1)}}. $

(2.4)

再由(2.3) 和(2.4) 式可知

$ {\rm const}.\geq \int_{\Omega}|{\rm grad}(|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i)|^{p_i}dx. $

(2.5)

因$p_i \geq 2, $故$\frac{p_i^2}{2(p_i-1)}\leq p_i.$因此$\{|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i\}$在$W^{1, \frac{p_i^2}{2(p_i-1)}} (\Omega)$中有界.利用引理1.4 $\{|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i\}$在$L^{p_i}(\Omega)$中相对紧.因为Nemytskii映射$u \in L^{p_i}(\Omega) \rightarrow |u_i|^{\frac{p_i}{2(p_i-1)}}{\rm sgn}u_i \in L^{p_i}(\Omega)$连续, 所以$\{u_i\}$在$L^{p_i}(\Omega)$中相对紧, $i = 1, 2, \cdots, M$.于是$\{u\}$在$Y$中相对紧.至此证明了$(I+A)^{-1}$为紧映射.

(ⅱ)$\frac{2N}{N+1} < p_i < 2.$

为此, 定义$\eta_{n}^{(i)}, \theta_{n}^{(i)}: R\rightarrow R$如下

$ \begin{eqnarray}\eta_{n}^{(i)}(t) = ~\left\{ \begin{array}{lll} |t|^{p_i-1}{\rm sgn}t, ~~\forall |t| \geq \frac{1}{n}, \\ (\frac{1}{n})^{p_i-2}t, ~~\forall |t| \leq \frac{1}{n}\\ \end{array} \right.\end{eqnarray} $

(2.6)

和

$ \begin{eqnarray}\theta_{n}^{(i)}(t) =~\left\{ \begin{array}{lll} |t|^{2-\frac{2}{p_i}}{\rm sgn}t, ~~\forall |t| \geq \frac{1}{n}, \\ (\frac{1}{n})^{1-\frac{2}{p_i}}t, ~\forall |t| \leq \frac{1}{n}.\\ \end{array} \right.\end{eqnarray} $

(2.7)

于是

$ [\eta_{n}^{(i)}(t)]' = (p_i-1)\times (\frac{p'_i}{2})^{p_i}\times \{[\theta_{n}^{(i)}(t)]'\}^{p_i}, ~~\forall |t|\geq \frac{1}{n} $

且$ [\eta_{n}^{(i)}(t)]' = \{[\theta_{n}^{(i)}(t)]'\}^{p_i}, ~~\forall |t|\leq \frac{1}{n}. $因为$\eta_{n}^{(i)}$单调、Lipschitz连续、$\eta_{n}^{(i)}(0) = 0 $且$(\eta_{n}^{(i)})'$在$R$上除至多有限点外连续, 所以引理2.4蕴含$\langle \eta_{n}^{(i)}(u_i), \partial\Phi_{i}(u_i)\rangle\geq 0, $ $u \in W^{1, p_i}(\Omega).$

下面计算

$ \langle|u_i|^{p_i-1}{\rm sgn}u_i, A_{i}u_i\rangle = \mathop {\lim }\limits_{n \to \infty }\langle \eta_{n}^{(i)}(u_i), A_{i}u_i\rangle\nonumber\\ \geq \mathop {\lim }\limits_{n \to \infty }\langle\eta_{n}^{(i)}(u_i), B_{i}u_i\rangle\nonumber\\ = \mathop {\lim }\limits_{n \to \infty }\int _{\Omega} (1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i}[(\eta_{n}^{(i)})'(u_i)]dx\nonumber\\ + \lambda_i \mathop {\lim }\limits_{n \to \infty } \int _{\Omega}|u_i|^{q_i-2}u_i \eta_{n}^{(i)}(u_i)dx+\lambda_i \mathop {\lim }\limits_{n \to \infty } \int _{\Omega}|u_i|^{r_i-2}u_i \eta_{n}^{(i)}(u_i)dx\nonumber\\ \geq \mathop {\lim }\limits_{n \to \infty }\int _{\Omega}|\nabla u_i|^{p_i}[(\eta_{n}^{(i)})'(u_i)]dx\nonumber\\ \geq {\rm const} \cdot \mathop {\lim }\limits_{n \to \infty }\int _{\Omega}|{\rm grad}(\theta_{n}^{(i)}(u_i))|^{p_i}dx\nonumber\\ \geq {\rm const} \int_{\Omega}|{\rm grad}(|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i)|^{p_i}dx. $

(2.8)

因$w_i = u_i + A_{i}u_i$, 故

$ \|w_i\|_{p_i} \||u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i\|_{\frac{{p_i}^{2}}{2(p_i-1)}}^{\frac{{p_i}^{2}}{2(p_i-1)p'_i}}\nonumber\\ \geq \langle|u_i|^{p_i-1}{\rm sgn}u_i, w_i\rangle\nonumber\\ = \langle |u_i|^{p_i-1}{\rm sgn}u_i, u_i\rangle +(|u_i|^{p_i-1}{\rm sgn}u_i, A_{i}u_i\rangle\nonumber\\ \geq \||u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i\|_{\frac{{p_i}^{2}}{2(p_i-1)}}^{\frac{{p_i}^{2}}{2(p_i-1)}}+ {\rm const} \cdot \|{\rm grad}(|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i)\|_{p_i}^{p_i}, $

(2.9)

于是由$\frac{2N}{N+1}< p_i < 2 $和(2.9) 式可知

$ \||u_i|^{2-\frac{2}{p_i}}{\rm sgn}u\|_{p_i}^{\frac {p_i}{2(p_i-1)}} \leq \||u_i| ^{2-\frac{2}{p_i}}{\rm sgn}u_i\|_{\frac{{p_i}^{2}}{2(p_i-1)}}^{\frac{p_i}{2(p_i-1)}} \leq \|w_i\|_{p_i}\leq {\rm const}. $

再利用(2.9) 式, $ \|{\rm grad}(|u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i)\|_{p_i}\leq {\rm const}.$从而$\{ |u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i \}$在$W^{1, p_i}(\Omega)$中有界.

由引理1.4知当$N\geq 2$时,

$ W^{1, p_i}(\Omega)\hookrightarrow\hookrightarrow L^{\frac{{p_i}^{2}}{2(p_i-1)}}(\Omega); $

当$N=1$时,

$ W^{1, p_i}(\Omega)\hookrightarrow\hookrightarrow C_{B}(\Omega), $

所以$\{ |u_i|^{2-\frac{2}{p_i}}{\rm sgn}u_i \}$在$ L^{\frac{{p_i}^{2}}{2(p_i-1)}}(\Omega)$中相对紧.又因为Nemytskii映射

$ u_i \in L^{\frac{{p_i}^{2}}{2(p_i-1)}}(\Omega) \rightarrow |u_i|^{\frac{p_i}{2(p_i-1)}}{\rm sgn}u \in L^{p_i}(\Omega) $

连续, 所以$\{u_i\}$在$L^{p_i}(\Omega)$中相对紧.于是$\{u\}$在$Y$中相对紧.至此证明了$(I+A)^{-1}$为紧映射.

命题2.3 定义$C : Y \rightarrow Y$如下

$ Cu = (C_1u_1, C_2 u_2, \cdots, C_Mu_M), $

$\forall u = (u_1, u_2, \cdots, u_M)\in Y, $其中$C_i : L^{p_i}(\Omega)\rightarrow L^{p_i}(\Omega)$为$C_iu_i = \varepsilon_i g_i(x, \nabla u_i, u_i), $ $i = 1, 2, \cdots, M.$则$C : Y \rightarrow Y$连续.

证 由$g_i$的假设条件可知$C_i: L^{p_i}(\Omega)\rightarrow L^{p_i}(\Omega)$有定义且$\forall u_i, v_i \in L^{p_i}(\Omega), $

$ \|C_iu_i-C_iv_i\|_{p_i}\leq b_i\varepsilon_i\|u_i-v_i\|_{p_i}, $

$i = 1, 2, \cdots, M.$由此可知$C_i$连续, $i = 1, 2, \cdots, M.$从而由乘积空间的性质$C : Y\rightarrow Y$连续.

命题2.4 映射$C(I+A)^{-1}: Y \rightarrow Y$为紧映射.

证 由命题2.2和2.3易知映射$C(I+A)^{-1}: Y \rightarrow Y$为紧映射.

定理2.1 若存在$u = (u_1, u_2, \cdots, u_M)\in Y$使得对$ f = (f_1, f_2, \cdots, f_M) \in Y$具$f \neq \theta$, 满足

$ \int_\Omega [g_i(x, \nabla u_i, u_i)-f_i]|u_i|^{p_i-2}u_i dx \geq 0, $

(2.10)

$i = 1, 2, \cdots, M, $这里$\theta$为$Y$中的零元, 则$u = (u_1, u_2, \cdots, u_M)\in Y$为Capillarity系统(2.1) 的非平凡解.

如果进一步假设$g_i(x, r_1, \cdots, r_{N+1})$关于$r_{N+1}$单调, 即

$ (g_i(x, r_1, \cdots, r_{N+1})-g_i(x, t_1, \cdots, t_{N+1}))(r_{N+1} - t_{N+1}) \geq 0 , $

$\forall x \in \Omega$及$(r_1, \cdots, r_{N+1}), (t_1, \cdots, t_{N+1})\in R^{N+1}, $ $i = 1, 2, \cdots, M, $那么Capillarity系统(2.1) 存在唯一非平凡解.

证 若$ f = (f_1, f_2, \cdots, f_M) \in Y$满足(2.10), 则由命题2.1-2.4知定理1.1的条件被满足.从而$ u = (u_1, u_2, \cdots, u_M) \in Y$满足算子方程$ f = Au + Cu$.

由引理2.3, 命题2.1和命题2.3知$A\theta + C\theta = \theta.$因此由$f \neq \theta$知$u \neq \theta.$因$f = Au +Cu, $故$f_i = B_{i} u_i + \partial \Phi_i(u_i)+C_iu_i, $ $i = 1, 2, \cdots, M.$ $\forall \psi \in C_0^{\infty}(\Omega), $利用引理2.1知

$ \langle \psi, f_i\rangle = \langle \psi, B_{_i}u_i + \partial \Phi_i(u_i)+C_iu_i \rangle = \langle \psi, B_{i}u_i+C_iu_i \rangle\\ \;\;\;\;\;\;= \int_\Omega <(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i, \nabla \psi> dx + \int_\Omega \varepsilon_ig_i(x, \nabla u_i, u_i)\psi dx\\ \;\;\;\;\;\;+ \lambda_i\int_\Omega |u_i|^{q_i-2}u_i \psi dx + \lambda_i \int_\Omega |u_i|^{r_i-2}u_i \psi dx\\ \;\;\;\;\;\; = \int_\Omega - {\rm div}[(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i] \psi dx \\ \;\;\;\;\;\;+ \lambda_i\int_\Omega |u_i|^{q_i-2}u_i \psi dx + \lambda_i \int_\Omega |u_i|^{r_i-2}u_i \psi dx+ \int_\Omega \varepsilon_ig_i(x, \nabla u_i, u_i)\psi dx, $

因此

$ -{\rm div}[(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2{p_i}}}})|\nabla u_i|^{p_i-2}\nabla u_i]+ \lambda_i(|u_i|^{q_i-2}u_i + |u_i|^{r_i-2}u_i)+ \varepsilon_ig_i(x, \nabla u_i, u_i)\\ = f_i(x)~{\rm a.e.}~~ x \in \Omega. $

下证$-< \vartheta, (1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i> \in \beta_{x}(u_i(x)), ~~ x \in \Gamma.$

利用Green公式, $ \forall v_i \in W^{1, p_i}(\Omega)$,

$ \int_{\Gamma}< \vartheta, (1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i>v_i|_{\Gamma}d\Gamma(x)\\ = \int_{\Omega}{\rm div}[(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i]v_i dx\\ + \int _{\Omega} <(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i, \nabla v_i>dx\\ = \int_{\Omega} \lambda (|u_i|^{q_i-2}u_i + |u_i|^{r_i-2}u_i)v_i dx + \int_{\Omega}\varepsilon_ig_i(x, \nabla u_i, u_i)v_i dx-\int_\Omega f_i v_i dx\\ + \int _{\Omega} <(1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i, \nabla v_i>dx \\ = - \int_\Gamma \beta_x(u_i|_\Gamma) v_i|_\Gamma d\Gamma(x). $

因此

$ -< \vartheta, (1+\frac{|\nabla u_i|^{p_i}}{\sqrt{1+|\nabla u_i|^{2p_i}}})|\nabla u_i|^{p_i-2}\nabla u_i> \in \beta_{x}(u_i(x)), ~~ x \in \Gamma. $

至此证明了$u = (u_1, u_2, \cdots, u_M)$是(2.1) 式的非平凡解.

最后证明若进一步假设$g_i(x, r_1, \cdots, r_{N+1})$关于$r_{N+1}$单调, 则(2.1) 式的解还是唯一的.

事实上, 只需证明若$f = Au + Cu = A\widehat{u}+C\widehat{u}, $其中$u = (u_1, u_2, \cdots, u_M)$, $\widehat{u}= (\widehat{u_1}, \widehat{u_2}, \cdots, \widehat{u_M})$, 则: $u \equiv \widehat{u}, $即$u_i \equiv \widehat{u_i}, $ $i = 1, 2, \cdots, M.$

在附加条件“$g_i(x, r_1, \cdots, r_{N+1})$关于$r_{N+1}$单调”的假设下, 易知$C_i$单调.利用引理2.1及2.2并注意下式

$0 \leq \langle u_i-\widehat{u_i}, B_{i} u_i - B_{i}\widehat{u_i}\rangle = \langle u_i-\widehat{u_i}, \partial \Phi_i (\widehat{u_i}) - \partial \Phi_i (u_i)\rangle + \langle u_i-\widehat{u_i}, C_i\widehat{u_i} - C_iu_i \rangle\leq 0, $

可知$\langle u_i-\widehat{u_i}, B_{i} u_i - B_{i}\widehat{u_i}\rangle = 0, $ $i = 1, 2, \cdots, M.$因此$u \equiv \widehat{u}.$

推论2.1 当$i \equiv 1$时, Capillarity系统退化成如下Capillarity方程

$ \begin{equation}\left\{ \begin{array}{lll} -{\rm div}[(1+\frac{|\nabla u|^{p}}{\sqrt{1+|\nabla u|^{2{p}}}})|\nabla u|^{p-2}\nabla u]+\\ \lambda(|u|^{q-2}u + |u|^{r-2}u) + \varepsilon g(x, \nabla u, u)= f(x) ~~{\rm a.e.}~~x\in \Omega, \\ - <\vartheta, (1+\frac{|\nabla u|^{p}}{\sqrt{1+|\nabla u|^{2{p}}}}) |\nabla u|^{p-2}\nabla u> \in \beta_x(u(x))~~{\rm a.e.}~~x\in \Gamma. \end{array} \right.\end{equation} $

(2.11)

当$f(x) \in L^p(\Omega)$满足$\displaystyle\int_\Omega [g(x, \nabla u, u)-f]|u|^{p-2}u dx \geq 0$时, (2.11) 式存在非平凡解$u(x) \in L^p(\Omega).$若进一步假设$g_i(x, r_1, \cdots, r_{N+1})$关于$r_{N+1}$单调, 则(2.11) 式存在唯一的非平凡解.

注2.1 文[1]在讨论Capillarity边值问题(1.1) 解的存在性时, 不仅要证明所定义的算子$A$和$L$是极大单调算子, 还需要验证一个很复杂的不等式“$\langle Aw, J^{-1}(L_t(w))\rangle \geq - k_1 \|L_tw\|^2 - k_2\|L_tw\| - k_3, $其中$L_t$为$L$的Yosida逼近”; 文[3]在讨论Capillarity边值问题(1.2) 解的存在性时, 不仅要验证$A$是$m$增生、有界逆紧映射并且满足条件“$\langle Au - f, J(u-a)\rangle \geq C(a, f), $其中$f \in R(A)$且$a \in D(A)$”, 还需要挖掘$A$的值域的特征.

本文采用了不同于文[1, 3]中的证明方法.从某种意义上讲, 研究方法相对简单.