该文研究以下$ k $-Hessian问题的$ k $凸径向正解的存在性和边界渐近行为

(1.1) $ \begin{equation} \left\{\begin{array}{ll} S_{k}\left(\lambda\left(D^{2} z\right)\right) = b(x) f(z), &\mbox{ in }\; \; \Omega, \\ z = +\infty, &\mbox{ on } \; \; \partial\Omega, \end{array}\right. \end{equation} $

其中, $ \Omega \subset {{\Bbb R}} ^{N} $$ (N \geq 2) $是一个严格凸的光滑有界区域, 方程(1.1) 中的条件表示当$ d(x) = dist(x, \partial\Omega) \rightarrow 0 $时, $ z(x) \rightarrow +\infty $. 若$ \Omega = B_{R} $, 其中, $ B_{R}\subset {{\Bbb R}} ^{N} $是以原点为心半径为$ R>0 $的球, 则方程(1.1) 的条件应替换为$ z(x) \rightarrow +\infty, \; (x \rightarrow R) $. 在$ B_{R} $上, 方程(1.1) 满足$ z(x) \rightarrow +\infty, \; (x \rightarrow R) $的解被称为爆破解. 特别地, 若$ \Omega = {{\Bbb R}} ^{N} $, 则解被称为整体爆破解.

$ S_{k}\left(\lambda\left(D^{2} z\right)\right) $是$ k $-Hessian算子形式如下

$ S_{k}\left(\lambda\left(D^{2} z\right)\right) = \sum\limits_{1\leq j_{1}<\cdots <j_{k}\leq N} \lambda_{j_{1}}\cdot \lambda_{j_{2}} \cdots \lambda_{j_{k}}, \; \; k = 1, 2, \cdots , N, $

其中, $ D^{2} z = \left(\frac{\partial^{2}z}{\partial x_{i}\partial x_{j}}\right) $是$ z \in C^{2}({{\Bbb R}} ^{N}) $的Hessian矩阵, $ \lambda\left(D^{2} z\right) = \left(\lambda_{1}, \lambda_{2}, \cdots , \lambda_{N}\right) $是Hessian矩阵$ D^{2} z $的特征值构成的向量, $ \lambda_{1}, \lambda_{2}, \cdots , \lambda_{N} $是Hessian矩阵$ D^{2} z $的特征值.

令锥$ \Gamma_{k} $为$ \{\lambda \in {{\Bbb R}} ^{N}:S_{k}\left(\lambda\left(D^{2} z\right)\right)>0\}\subset {{\Bbb R}} ^{N} $且含正锥

$ \Gamma^{+} = \{\lambda \in {{\Bbb R}} ^{N}:\lambda_{i}>0, i = 1, 2, \cdots , N\}, $

则由文献[1] 和[2]可知

$ \Gamma^{+} = \Gamma_{N}\subset \cdots \subset\Gamma_{k+1}\subset\Gamma_{k}\subset \cdots \subset\Gamma_{1}. $

定义 1.1^[3]   令$ \Omega $是$ {{\Bbb R}} ^{N} $的有界开子集且$ k \in \{1, 2, \cdots , N\} $. 若对于$ x \in \Omega $, $ (\lambda_{1}, \lambda_{2}, \cdots , $$ \lambda_{N}) \in \overline{\Gamma}_{k} $, 则函数$ z \in C^{2}(\Omega) $是$ k $凸. 换句话说, 若在$ \Omega $中, $ S_{i}\left(\lambda\left(D^{2} z\right)\right) \geq 0 $$ (i = 1, 2, \cdots k) $, 则函数$ z \in C^{2}(\Omega) $是$ k $凸.

定义 1.2^[3]   令$ \Omega \subset {{\Bbb R}} ^{N} $是具有$ C^{2} $边界的开集且$ k \in \{1, 2, \cdots , N-1\} $. 若对于$ \bar{x} \in \partial\Omega $, $ S_{i}(\kappa_{j}(\bar{x}), \cdots , \kappa_{N-1}(\bar{x})) > 0 $$ (i = 1, 2, \cdots , k) $, 则$ \Omega $是严格凸的. 其中, $ \kappa_{j}(\bar{x}) $$ (j = 1, 2, \cdots , N-1) $是$ \partial\Omega $关于$ \bar{x} $的主曲率.

$ S_{k}\left(\lambda\left(D^{2} z\right)\right) $是一个包含诸多著名算子的集合. 当$ k = 1 $时, $ k $-Hessian算子是Laplace算子; 当$ k = N $时, $ k $-Hessian算子是Monge-Ampère算子, 即

近年来, Laplace问题和Monge-Ampère问题被广泛应用于数学和应用数学的诸多分支. 关于Monge-Ampère问题和Laplace问题解的存在性、非存在性、多解性、唯一性和渐近稳定性结果见文献[4-17]. 例如, 在2015年, 应用Karamata正则变化理论(其被Karamata于1930年建立, 是研究随机过程^[18-20]和非线性椭圆问题解的渐近稳定性的一个有用的工具), 张^[4]研究了下列边界爆破Monge-Ampère问题严格凸解的边界行为

(1.2) $ \begin{equation} \left\{\begin{array}{ll} {\rm det} D^{2} z = b(x) f(z), &\mbox{ in }\; \; \Omega, \\ z = +\infty, &\mbox{ on } \; \; \partial\Omega, \end{array}\right. \end{equation} $

其中, $ \Omega \subset {{\Bbb R}} ^{N} $是一个严格凸的光滑有界区域.

在2019年, 应用Karamata正则变化理论, Khamessi和Othman^[6]研究了下列奇异Laplace问题正解的边界渐近行为

(1.3) $ \begin{equation} \left\{\begin{array}{ll} \triangle z = b(x) f(z), &\mbox{ in }\; \; \Omega, \\ z>0, &\mbox{ in }\; \; \Omega, \\ z = +\infty, &\mbox{ on } \; \; \partial\Omega, \end{array}\right. \end{equation} $

其中, $ \Omega \subset {{\Bbb R}} ^{N} $是一个严格凸的光滑有界区域.

在2019年, 应用单调迭代方法, Covei^[7]分别研究了下列Laplace方程整体有界径向解和爆破径向解的存在性

(1.4) $ \begin{equation} \triangle z = b(|x|) f(z)+h(|x|) g(z), \; \; x \in {{\Bbb R}} ^{N}, \end{equation} $

在方程(1.4) 中, 当$ f(z) = z^{\alpha} $, $ g(z) = z^{\beta} $, $ 0<\alpha \leq \beta $时, Lair^[8]研究了整体爆破径向正解的存在性和非存在性.

在2012年, Dupaigne等^[9]分别得到了下列半线性椭圆方程整体有界解和爆破解的存在性, 并且研究了解的唯一性、对称性和渐近稳定性

(1.5) $ \begin{equation} \triangle z = b(|x|) f(z), \; \; x \in {{\Bbb R}} ^{N}. \end{equation} $

在问题(1.1) 中, 当$ k = 1 $时, 这些问题源于现实世界的许多现象. 例如, 生产规划问题相应的标量情况模型^[21]. 当$ k \geq 2 $时, $ k $-Hessian方程是完全非线性偏微分方程. 在流体力学、几何问题和其它应用学科中, 它有着重要的应用. 例如, 在文献[17]中, $ k $-Hessian问题可以描述反射器的形状设计或Weingarten曲率. 近年来, 诸多学者对$ k $-Hessian问题产生了浓厚的研究兴趣, 并且应用诸如单调迭代方法、上下解方法、不动点定理、变分法、移动平面法和Perron's方法等方法得到了很多关于$ k $-Hessian问题的优秀成果, 参见文献[22-34].

在2015年, 应用Arzela-Ascoli定理和单调迭代方法, 张和周^[22]分别得到了$ k $-Hessian方程

(1.6) $ \begin{equation} S_{k}\left(\lambda\left(D^{2} z\right)\right) = b(|x|) f(z), \; \; x \in {{\Bbb R}} ^{N} \end{equation} $

和Hessian方程组

(1.7) $ \begin{equation} \left\{\begin{array}{ll}{ S_{k}\left(\lambda\left(D^{2} z_{1}\right)\right) = b(|x|) f(z_{2}), \; \; x \in {{\Bbb R}} ^{N}}, \\ { S_{k}\left(\lambda\left(D^{2} z_{2}\right)\right) = h(|x|) g(z_{1}), \; \; x \in {{\Bbb R}} ^{N}}\end{array}\right. \end{equation} $

径向解的有界性和爆破性.

在2017年, 应用Perron's方法, 曹和保^[34]研究了Hessian方程Dirichlet问题粘性解的存在性和唯一性

(1.8) $ \begin{equation} S_{k}\left(\lambda\left(D^{2} z\right)\right) = f(x), \; \; x \in {{\Bbb R}} ^{N}\setminus \overline{\Omega}, \end{equation} $

其中, $ \Omega \subset {{\Bbb R}} ^{N} $$ (N \geq 3) $是一个光滑有界区域.

在2019年, 应用上下解方法, 马和李^[26]研究了$ k $-Hessian方程(1.1) 粘性解的存在性和渐近稳定性. 这里, $ f $满足Keller-Osserman型条件

(1.9) $ \begin{equation} \int_{t}^{\infty} \frac{1}{((k+1) F_{1}(s))^{\frac{1}{k+1}}} {\rm d}s<\infty, \quad t>0, \end{equation} $

其中, $ F_{1}(s) = \int_{0}^{s} f(\tau){\rm d}\tau $.

在2020年, 应用上下解方法, 冯和张^[27]得到了下列边界爆破问题$ k $凸径向解的存在性与非存在性

(1.10) $ \begin{equation} S_{k}\left(\lambda\left(D^{2} z\right)\right) = b(x) [z(x)]^{k} [\ln z(x)]^{\beta}, \; \; x \in \Omega, \; \; z\mid_{\partial\Omega} = +\infty, \end{equation} $

其中, $ \beta>0 $.

受文献[15] 的启发, 本文首先利用单调迭代方法, 分别在球$ B_{R} $和全空间$ {{\Bbb R}} ^{N} $上, 研究了$ k $-Hessian方程(1.1) 严格凸的径向对称正解的存在性. 然后应用上下解方法和Karamata正则变化理论, 在严格凸的光滑有界区域$ \Omega $上, 得到了$ k $-Hessian方程(1.1) 严格凸的爆破解的边界渐近行为. 关于单调迭代方法和上下解方法在研究非线性问题方面的作用, 请参见文献[22-27, 35-43]. 我们的主要结果改进和扩展了以前的工作, 请参见文献[1, 3-4, 9, 15-16, 22, 26].

该节给出一些预备知识和关于Karamata正则变化理论的著名结果. 记$ \psi $是

(2.1) $ \begin{equation} \int_{\psi(t)}^{\infty} \frac{{\rm d}s}{(k f(s))^{\frac{1}{k}}} = t, \quad \forall t>0 \end{equation} $

的唯一解. 设$ f $满足

$ (S_{1}) $   $ f \in C^{1}[0, \infty) $是一个严格单调递增的函数且$ f(0) = 0 $;

$ (S_{2}) $   $ \int_{s}^{\infty} \frac{{\rm d}t}{F(t)}<\infty $, $ F(s): = (k f(s))^{\frac{1}{k}} $, $ s>0 $;

$ (S_{3}) $   存在一个正常数$ C^{+}_{f} $使得

$ \lim\limits _{s \rightarrow +\infty}J_{f}(s) = C^{+}_{f}, $

其中, $ J_{f}(s): = F^{\prime}(s) \int_{s}^{\infty} \frac{{\rm d}t}{F(t)}, s>0 $.

注 2.1  (1) 若$ (S_{1}) $和$ (S_{2}) $成立, 则$ f $满足Keller-Osserman型条件^{[1, 3-4, 16, 26]} (1.9) 式. 详细的证明请参看文献[15];

(2) 由$ (S_{1}) $, $ (S_{2}) $和$ (S_{3}) $可知$ \psi(t) \rightarrow +\infty \Leftrightarrow t \rightarrow 0. $

令$ b $满足

$ (B_{1}) $   在$ \Omega $中, $ b \in C^{\infty}(\Omega) $是一个正函数.

$ (B_{2}) $   $ b $是一个球对称连续函数, 即, 对于$ r = |x| $, $ b(x) = b(r) $.

$ (B_{3}) $   存在$ g \in \Lambda $和正常数$ b_{i} (i = 1, 2) $使得

$ b_{1}: = \lim\limits _{d(x) \rightarrow 0} \inf \frac{b(x)}{g^{k+1}(d(x))} \leq b_{2}: = \lim\limits _{d(x) \rightarrow 0} \sup \frac{b(x)}{g^{k+1}(d(x))}, $

其中, $ \Lambda $是一个所有单调正函数$ g \in C^{1}(0, \delta_{0}) \bigcap L^{1}(0, \delta_{0}) $$ (\delta_{0} > 0) $的集合且$ g $满足

(2.2) $ \begin{equation} \lim\limits _{t \rightarrow 0^{+}} \frac{\rm d}{{\rm d}t}\left(\frac{G(t)}{g(t)}\right): = G_{g} \in[0, \infty), \quad G(t): = \int_{0}^{t} g(v) {\rm d}v. \end{equation} $

注 2.2^[15]   关于$ g \in \Lambda $的一些函数列举如下:

(1) 对于$ \mu > 0 $, 当$ g(t) = \exp(-\frac{1}{t^{\mu}}) $, $ t \in (0, \delta_{0}) $时, $ G_{g} = 0 $;

(2) 对于$ \mu > 0 $, 当$ g(t) = t^{\mu} $, $ t \in (0, \delta_{0}) $时, $ G(t) = \frac{t^{1+\mu}}{1+\mu} $, $ G_{g} = \frac{1}{1+\mu} $;

(3) 对于$ \mu > 0 $, 当$ g(t) = \frac{1}{(-\ln t)^{\mu}} $, $ t \in (0, 1) $时, $ G_{g} = 1 $;

(4) 对于$ \mu > 0 $, 当$ g(t) = \frac{1}{t^{\mu}} $, $ t \in (0, \delta_{0}) $时, $ G(t) = \frac{t^{1-\mu}}{1-\mu} $, $ G_{g} = \frac{1}{1-\mu} $.

关于$ g \in \Lambda $的一些性质列举如下.

引理 2.1^[4]   如果$ g \in \Lambda $, 那么

(1) $ \lim\limits _{t \rightarrow 0^{+}} \frac{G(t)}{g(t)} = 0 $;

(2) $ \lim\limits _{t \rightarrow 0^{+}} \frac{G(t)g^{\prime}(t)}{g^{2}(t)} = 1-\lim\limits _{t \rightarrow 0^{+}} \frac{\rm d}{{\rm d}t}\left(\frac{G(t)}{g(t)}\right) = 1-G_{g} $;

(3) 若$ G_{g} = 0 $, 则$ g $在$ 0 $处比$ t^{p} $增长的快, 其中, $ p > 1 $;

(4) 若$ G_{g}>0 $, 则$ g $在$ 0 $处是一个指数为$ \frac{1-G_{g}}{G_{g}} $的标准正则变化函数. 特别地, 若$ G_{g} = 1 $, 则$ g $在$ 0 $处是一个标准慢变函数.

为了研究满足条件$ (S_{2}) $和条件$ (S_{3}) $的$ f $以及(2.1) 式中$ \psi $的性质, 接下来, 引入关于Karamata正则变化理论的结果.

定义 2.1^[44]   如果对于$ \beta > 0 $和某个$ q \in {{\Bbb R}} $, 正函数$ f \in C^{1}[\zeta, \infty) $, $ \zeta > 0 $, 满足

(2.3) $ \begin{equation} \lim \limits_{s \rightarrow \infty} \frac{f(\beta s)}{f(s)} = \beta^{q}. \end{equation} $

那么$ f $在无穷远处是一个指数为$ q $的正则变化函数, 记$ f \in RV_{q} $. 特别地, 若$ q = 0 $, 则$ f $在无穷远处是一个慢变函数.

显然, 当$ f \in RV_{q} $时, $ L(s) : = \frac{f(s)}{s^{q}} $在无穷远处是慢变函数.

注 2.3  关于在无穷远处慢变函数的一些基本例子如下

(1) 每一个$ f(s) \in C[\zeta, \infty) $在无穷远处有正极限;

(2) $ f(s) = \exp((\ln s)^{\mu}) $, $ \mu \in (0, 1) $;

(3) $ f(s) = (\ln(\ln s))^{\mu} $, $ \mu \in {{\Bbb R}} $;

(4) $ f(s) = (\ln s)^{\mu} $, $ \mu \in {{\Bbb R}} $.

命题 2.1^[44](一致收敛定理)   当$ f \in RV_{q} $时, 对于$ \beta \in [\beta_{1}, \beta_{2}] $且$ 0 < \beta_{1} < \beta_{2} $, (2.3)式一致成立.

命题 2.2^[44](Karamata表达式定理)   $ H $在无穷远处是一个慢变函数当且仅当它可表示为以下形式

(2.4) $ \begin{equation} H(s) = z(s)\exp\left(\int_{\zeta_{1}}^{s} \frac{y(\tau)}{\tau}{\rm d}\tau \right), \quad s \geq \zeta_{1}, \end{equation} $

其中, $ \zeta_{1} \geq \zeta $, $ z $和$ y $是连续的, $ \lim\limits _{s \rightarrow \infty}y(s) = 0 $, $ \lim\limits _{s \rightarrow \infty}z(s) = c_{0} $, $ c_{0} > 0 $.

定义 2.2^[44]   (1)

(2.5) $ \begin{equation} \widetilde{H}(s) = c_{0} \exp\left(\int_{\zeta_{1}}^{s} \frac{y(\tau)}{\tau}{\rm d}\tau \right), \quad s \geq \zeta_{1}, \end{equation} $

在无穷远处是一个标准慢变函数;

(2)

(2.6) $ \begin{equation} f(s) = s^{q} \widetilde{H}(s), \quad s \geq \zeta_{1}, \end{equation} $

在无穷远处是一个指数为$ q $的标准正则变化函数(记$ f \in NRV_{q} $).

等价地, $ f \in RV_{q} $属于$ NRV_{q} $当且仅当对于$ \zeta_{1} > 0 $, $ f\in C^{1}[\zeta_{1}, \infty) $满足

(2.7) $ \begin{equation} \lim\limits _{s \rightarrow \infty} \frac{sf'(s)}{f(s)} = q. \end{equation} $

命题 2.3^[44]   若$ H $, $ H_{1} $在无穷远处是慢变函数, 则

(1) $ H^{q} $ ($ \forall q \in {{\Bbb R}} $), $ c_{1}H + c_{2}H_{1} $, $ H \cdot H_{1} $和$ H \circ H_{1} $ ($ \lim\limits _{s \rightarrow \infty} H_{1}(s) = \infty $) 在无穷远处是慢变函数, 其中, $ c_{1} \geq 0 $, $ c_{2} \geq 0 $, $ c_{1} + c_{2} > 0 $;

(2) $ \lim\limits _{s \rightarrow \infty} s^{q}H(s) \rightarrow \infty $, $ \lim\limits _{s \rightarrow \infty} \frac{H(s)}{s^{q}} \rightarrow 0 $, $ \forall q > 0 $;

(3) $ \lim\limits _{s \rightarrow \infty} \frac{\ln(H(s))}{\ln s} \rightarrow 0 $, $ \lim\limits _{s \rightarrow \infty} \frac{\ln(s^{q} H(s))}{\ln s} \rightarrow q $, $ \forall q \in {{\Bbb R}} $.

命题 2.4^[45](渐近行为)   若$ H $在无穷远处是一个慢变函数, 则对于$ a \geq 0 $, 当$ t \rightarrow \infty $时

(1) $ \int_{a}^{t} s^{q} H(s){\rm d}s\cong \frac{t^{q+1}}{q+1}, \quad q>-1 $;

(2) $ \int_{t}^{\infty} s^{q} H(s){\rm d}s\cong \frac{t^{q+1}}{-q-1}, \quad q<-1 $.

类似地, 对于正函数$ f \in C^{1}(-\infty, -\zeta) $, $ \zeta > 0 $ (或$ f \in C^{1}(0, s_{0}) $, $ s_{0} > 0 $), 可给出在$ -\infty $处(或在$ 0 $处)标准正则变化(或正则变化) 的定义和一些基本性质.

引理 2.2^[15]   假设$ f $满足$ (S_{1}) $和$ (S_{2}) $, 那么

(1) 当$ (S_{3}) $成立时, $ C^{+}_{f} \in [1, \infty) $;

(2) 当$ (S_{3}) $成立时, 存在$ T_{0} > 0 $使得在$ [T_{0}, \infty) $上$ \frac{f(s)}{s^{kl}} $是单调递增的, 其中, 对于$ C^{+}_{f} > 1 $, $ l \in \left(1, \frac{C^{+}_{f}}{C^{+}_{f} -1}\right) $; 对于$ C^{+}_{f} = 1 $, $ l \in (1, \infty) $;

(3) 对于$ C^{+}_{f} >1 $, $ (S_{3}) $成立当且仅当对于某个$ p > 1 $, $ f \in NRV_{kp} $. 事实上, 这种情况下, $ p = \frac{C^{+}_{f}}{C^{+}_{f} -1} $;

(4) 对于$ C^{+}_{f} = 1 $, 当$ (S_{3}) $成立时, 在无穷远处$ s^{p} $$ (p > k) $比$ f $增长的慢;

(5) 对于某个大的$ T_{0} > 0 $, 当$ f \in C^{2}(T_{0}, \infty) $且$ \lim\limits _{s \rightarrow +\infty} \frac{f(s)f''(s)}{f'^{2}(s)} = 1 $时, $ f $满足$ (S_{3}) $且$ C^{+}_{f} = 1 $.

记

(2.8) $ \begin{equation} \Upsilon(s): = sF'(\psi(s)) = \frac{sf'(\psi(s))}{(kf(\psi(s)))^{(k-1)/k}}, \quad s>0. \end{equation} $

引理 2.3^[15]   假设$ f $满足$ (S_{1}) $, $ (S_{2}) $和$ (S_{3}) $, 那么

(1) $ \psi(s)>0 $, $ s\in(0, \infty) $, $ \psi(0): = \lim\limits_{s\rightarrow 0^{+}}\psi(s) = +\infty $;

(2) $ -\psi'(s) = F(\psi(s)) = \sqrt[k]{kf(\psi(s))}, \quad s\in(0, \infty) $;

(3) $ \psi''(s) = F(\psi(s))F'(\psi(s)) = \frac{f'(\psi(s))}{(kf(\psi(s)))^{(k-2)/k}}, \quad s\in(0, \infty) $;

(4) $ \lim\limits _{s \rightarrow 0^{+}} \frac{s \psi^{\prime}(s)}{\psi(s)} = -\lim\limits _{s \rightarrow 0^{+}} \frac{s F(\psi(s))}{\psi(s)} = -\lim\limits _{t \rightarrow +\infty} \frac{F(t) \int_{t}^{\infty} \frac{{\rm d}\tau}{F(\tau)}}{t} = -\left(C^{+}_{f}-1\right) $;

(5) $ \lim\limits _{s \rightarrow 0^{+}} \frac{s \psi^{\prime \prime}(s)}{\psi^{\prime}(s)} = -\lim\limits _{s \rightarrow 0^{+}} \frac{s f^{\prime}(\psi(s))}{(k f(\psi(s)))^{(k-1) / k}} = -\lim\limits _{t \rightarrow +\infty} F^{\prime}(t) \int_{t}^{\infty} \frac{{\rm d}\tau}{F(\tau)} = -C^{+}_{f} $;

(6) 对于$ \beta \in [\beta_{1}, \beta_{2}] $, $ 0 < \beta_{1} < \beta_{2} $, $ \lim\limits _{s \rightarrow 0}\Upsilon(\beta s) = C^{+}_{f} $一致成立.

该节研究在球$ B_{R} $和全空间$ {{\Bbb R}} ^{N} $上$ k $-Hessian方程(1.1)径向对称正解的存在性.

引理 3.1^[46]   假设$ y(r) \in C^{2}[0, R), \; y^{\prime}(0) = 0 $. 那么, $ z(|x|) = y(r) \in C^{2}\left(B_{R}\right) $, $ r = |x|<R $, 并且

(3.1) $ \begin{equation} \lambda\left(D^{2} z\right) = \left\{\begin{array}{ll} { } \left(y^{\prime \prime}(r), \frac{y^{\prime}(r)}{r}, \cdots , \frac{y^{\prime}(r)}{r}\right), \quad &r \in(0, R), \\ { } \left(y^{\prime \prime}(0), y^{\prime \prime}(0), \cdots , y^{\prime \prime}(0)\right), \quad& r = 0, \end{array}\right. \end{equation} $

(3.2) $ \begin{equation} S_{k}\left(\lambda\left(D^{2} z\right)\right) = \left\{\begin{array}{ll} { } C_{N-1}^{k-1} y^{\prime \prime}(r)\left(\frac{y^{\prime}(r)}{r}\right)^{k-1}+C_{N-1}^{k} \left(\frac{y^{\prime}(r)}{r}\right)^{k}, & r \in(0, R), \\ C_{N}^{k}\left(y^{\prime \prime}(0)\right)^{k}, \quad& r = 0, \end{array}\right. \end{equation} $

其中$ B_{R} = \left\{x \in {{\Bbb R}} ^{N} :|x|<R\right\} $, $ C_{N}^{k} = \frac{N !}{k !(N-k) !} $.

通过计算, 可以得到以下引理:

引理 3.2   令$ \Omega = B_{R} $, $ z(|x|) = y(r) $是$ k $-Hessian方程(1.1) 的径向解当且仅当$ y(r) $是以下常微分方程的解

(3.3) $ \begin{equation} \left\{\frac{r^{N-k}}{k}\left[\left(y(r)\right)^{\prime}\right]^{k}\right\}^{\prime} = \frac{r^{N-1}}{C_{N-1}^{k-1}} b(r)f(y(r)), \quad r\in[0, R). \end{equation} $

注 3.1  令$ \Omega = B_{R} $, 且$ b \in C(B_{R}) $是正函数. 那么

(3.4) $ \begin{equation} \int_{0}^{R} \left( \int_{0}^{t} b(s) {\rm d}s\right)^{\frac{1}{k}}{\rm d}t = \infty \end{equation} $

当且仅当

(3.5) $ \begin{equation} \int_{0}^{R}\left(\frac{k}{t^{N-k}} \int_{0}^{t} \frac{s^{N-1}}{C_{N-1}^{k-1}}b(s) {\rm d}s\right)^{\frac{1}{k}} {\rm d}t = \infty. \end{equation} $

事实上, 由洛必达法则, 得

$ \lim\limits_{r\rightarrow R^{-}} \frac{ \int_{0}^{r} b(s) {\rm d}s}{\frac{k}{r^{N-k}} \int_{0}^{r} \frac{s^{N-1}}{C_{N-1}^{k-1}}b(s) {\rm d}s} = \frac{R^{1-k}C_{N-1}^{k-1}}{k} $

和

$ \begin{eqnarray*} \lim\limits_{r\rightarrow R^{-}} \frac{\int_{0}^{r} \left(\int_{0}^{t} b(s) {\rm d}s\right)^{\frac{1}{k}}{\rm d}t}{\int_{0}^{r}\left(\frac{k}{t^{N-k}} \int_{0}^{t} \frac{s^{N-1}}{C_{N-1}^{k-1}}b(s) {\rm d}s\right)^{\frac{1}{k}} {\rm d}t} = \lim\limits_{r\rightarrow R^{-}} \frac{ \left(\int_{0}^{r} b(s) {\rm d}s\right)^{\frac{1}{k}}}{\left(\frac{k}{r^{N-k}} \int_{0}^{r} \frac{s^{N-1}}{C_{N-1}^{k-1}}b(s) {\rm d}s\right)^{\frac{1}{k}}} = \left(\frac{R^{1-k}C_{N-1}^{k-1}}{k}\right)^{\frac{1}{k}}. \end{eqnarray*} $

由注3.1可知条件(3.4) 优于条件(3.5). 基于条件(3.4), 有以下两定理成立, 证明过程类似于文献[22] 中的定理1.1.

定理 3.1  令$ \Omega = B_{R} $. 假设

(1) $ f $满足$ (S_{1}) $和

$ \int_{1}^{\infty} \frac{{\rm d} s}{(f(s))^{\frac{1}{k}}} = \infty; $

(2) $ b $满足$ (B_{1}) $, $ (B_{2}) $和

$ \int_{0}^{R} \left(\int_{0}^{t} b(s) {\rm d}s\right)^{\frac{1}{k}}{\rm d}t = \infty. $

则方程(1.1) 有无穷多严格凸的径向正解.

定理 3.2  令$ \Omega = {{\Bbb R}} ^{N} $. 假设

(1) $ f $满足$ (S_{1}) $和

$ \int_{1}^{\infty} \frac{{\rm d}s}{(f(s))^{\frac{1}{k}}} = \infty; $

(2) $ b $满足$ (B_{1}) $, $ (B_{2}) $和

$ \int_{0}^{\infty} \left(\int_{0}^{t} b(s) {\rm d}s\right)^{\frac{1}{k}}{\rm d}t = \infty. $

则方程(1.1) 有无穷多严格凸的整体径向正解.

该节研究在严格凸的光滑有界区域$ \Omega $上$ k $-Hessian问题(1.1) 严格凸正解的渐近稳定性. 首先引入一些与$ k $-Hessian问题相关的引理.

引理 4.1^[24](比较原理)   对$ x \in \Omega $和$ z $, 定义一个函数$ f(x, z) $, 其中, $ z $属于某个包含$ z_{1} $和$ z_{2} $的区间, $ z_{1}, z_{2} \in C^{2}(\Omega)\bigcap C(\overline{\Omega}) $, $ \Omega \subset {{\Bbb R}} ^{N} $$ (N\geq2) $是一个有界区域. 若

(1) 对于$ \forall x \in \Omega $, $ f(x, z) $关于$ z $是一个严格单调递增的函数;

(2) 在$ \Omega $中, $ D^{2} z_{1} $是一个正定矩阵;

(3) $ S_{k}\left(\lambda\left(D^{2} z_{1}\right)\right)\geq f(x, z_{1}), \forall x \in \Omega $;

(4) $ S_{k}\left(\lambda\left(D^{2} z_{2}\right)\right)\leq f(x, z_{2}), \forall x \in \Omega $;

(5) $ z_{1}\leq z_{2}, \forall x \in \partial\Omega $.

则在$ \Omega $中, $ z_{1}\leq z_{2} $.

引理 4.2^[25]   令$ z \in C^{2}(\Omega) $使得对于任意的$ x \in \Omega $, $ (z_{x_{i}x_{j}}) $的所有主子矩阵是可逆的. 设$ h $是一个$ C^{2} $函数, 那么

(4.1) $ \begin{equation} S_{k}\left(\lambda\left(D^{2} h(z)\right)\right) = S_{k}\left(\lambda\left(D^{2} z\right)\right)\left[h^{\prime}(z)\right]^{k}+\left[h^{\prime}(z)\right]^{k-1} h^{\prime \prime}(z) \sum\limits_{i = 1}^{C_{N}^{k}} {\rm det} (z_{x_{is} x_{ij}})\left(\nabla z_{i}\right)^{T} B\left(z_{i}\right) \nabla z_{i}, \end{equation} $

其中, $ A^{T} $是矩阵$ A $的转置, $ B(z_{i}) $是$ i $阶主子矩阵$ (z_{x_{is}x_{ij}}) $的逆, $ \det(z_{x_{is}x_{ij}}) $是$ (z_{x_{is}x_{ij}}) $的行列式,

$ \nabla z_{i} = \left(z_{x_{i 1}}, z_{x_{i 2}}, \cdots, z_{x_{i k}}\right)^{T}, i = 1, 2, \cdots , C_{N}^{k}, $

其中$ C_{N}^{k} = \frac{N !}{k !(N-k) !} $.

对于$ \delta > 0 $, 令

$ \Omega_{\delta} = \{x\in\Omega:0<d(x)<\delta\}. $

若$ \Omega $是$ C^{l} $光滑的且$ l \geq 2 $, 则由文献[47] 中的引理14.16和引理14.17可知存在$ \delta_{1} \in (0, \delta_{0}) $使得$ d \in C^{l}(\Omega_{\delta_{1}}) $, 其中, $ \delta_{0} $在$ \Lambda $中已经被给定. 此外, 由文献[47] 中的引理14.17可知

(4.2) $ \begin{equation} \left\{\begin{array}{l} D d(x) = (0, 0, \cdots , 1), \\ D^{2} d(x) = {\rm diag} \left[-\varepsilon_{1}, \cdots , -\varepsilon_{N-1}, 0\right], \end{array}\right. \end{equation} $

其中, $ \varepsilon_{j} = \frac{\kappa_{j}\left(\bar{x}\right)}{1-\kappa_{j}\left(\bar{x}\right) d\left(x\right)} $, $ \kappa_{j}(\bar{x}) $$ (j = 1, 2, \cdots , N-1) $是$ \partial\Omega $关于$ \bar{x} $的主曲率, $ \bar{x} $是点$ x \in \Omega_{\delta_{1}} $到$ \partial\Omega $的投影.

引理 4.3^[48]   若在$ (0, \delta_{1}) $上, $ h $是一个$ C^{2} $函数. 则, 对于$ x \in \Omega_{\delta_{1}} $, 有

(4.3) $ \begin{eqnarray} S_{k}\left(\lambda\left(D^{2} h\left(d(x)\right)\right)\right) & = &\left[-h^{\prime}\left(d\left(x\right)\right)\right]^{k} S_{k}\left(\varepsilon_{1}, \cdots , \varepsilon_{N-1}\right){}\\ &&+\left[-h^{\prime}\left(d\left(x\right)\right)\right]^{k-1} h^{\prime \prime}\left(d\left(x\right)\right) S_{k-1}\left(\varepsilon_{1}, \cdots , \varepsilon_{N-1}\right). \end{eqnarray} $

定理 4.1  令$ f $满足$ (S_{1}) $, $ (S_{2}) $和$ (S_{3}) $, $ b $满足$ (B_{1}) $和$ (B_{3}) $. 如果

(4.4) $ \begin{equation} C^{+}_{f}>1, \end{equation} $

或

(4.5) $ \begin{equation} C^{+}_{f} = 1 \quad \mbox{ 且 } \quad G_{g}>0, \end{equation} $

那么$ k $-Hessian问题(1.1) 的每一个严格凸解$ z $满足

(4.6) $ \begin{equation} 1 \leq \lim\limits_{d(x) \rightarrow 0} \inf \frac{z(x)}{\psi\left(\xi_{2} G^{\beta}(d(x))\right)} \quad \mbox{ 和 } \quad \lim\limits_{d(x) \rightarrow 0} \sup \frac{z(x)}{\psi\left(\xi_{1} G^{\beta}(d(x))\right)} \leq 1, \end{equation} $

其中, $ \beta = \frac{k+1}{k} $, $ \psi $是方程(2.1) 的解, 且

$ \xi_{1} = \frac{k}{k+1}\left(\frac{b_{1}}{M_{k-1}\left(k G_{g}+\left(C^{+}_{f}-1\right)(k+1)\right)}\right)^{1 / k}, $

$ \xi_{2} = \frac{k}{k+1}\left(\frac{b_{2}}{m_{k-1}\left(k G_{g}+\left(C^{+}_{f}-1\right)(k+1)\right)}\right)^{1 / k}, $

这里

$ \begin{eqnarray*} M_{k-1}: = \max _{\bar{x} \in \partial \Omega} S_{k-1}(\kappa_{1}(\bar{x}), \cdots , \kappa_{N-1}(\bar{x})) \mbox{ 和 } m_{k-1}: = \min _{\bar{x} \in \partial \Omega} S_{k-1}(\kappa_{1}(\bar{x}), \cdots , \kappa_{N-1}(\bar{x})). \end{eqnarray*} $

证   令

(4.7) $ \begin{equation} \omega_{1}(d(x)) = G^{\beta}(d(x)), \quad \beta = \frac{k+1}{k}, \quad x \in \Omega_{\delta_{1}}. \end{equation} $

(4.8) $ \begin{equation} M_{k-1}: = \max\limits _{\bar{x} \in \partial \Omega} S_{k-1}(\kappa_{1}(\bar{x}), \cdots , \kappa_{N-1}(\bar{x}))   \mbox{ 和 }   m_{k-1}: = \min\limits _{\bar{x} \in \partial \Omega} S_{k-1}(\kappa_{1}(\bar{x}), \cdots , \kappa_{N-1}(\bar{x})). \end{equation} $

固定$ \epsilon \in (0, \min\{\frac{1}{2}, \frac{b_{1}}{2}\}) $, 选取

(4.9) $ \begin{equation} \begin{array}{ll} { }\xi_{1-} = \frac{k}{k+1}\left(\frac{(b_{1}-\epsilon)(1-\epsilon)-\epsilon}{M_{k-1}\left(k G_{g}+\left(C^{+}_{f}-1\right)(k+1)\right)}\right)^{1 / k}, \\ { }\xi_{2+} = \frac{k}{k+1}\left(\frac{(b_{2}+\epsilon)(1+\epsilon)+\epsilon}{m_{k-1}\left(k G_{g}+\left(C^{+}_{f}-1\right)(k+1)\right)}\right)^{1 / k}, \end{array} \end{equation} $

其中, $ b_{1} $和$ b_{2} $在条件$ (B_{3}) $中已经被给定. 利用引理2.1, 引理2.3, 有

$ G \in C^{2}(0, \delta_{0}) \bigcap C[0, \delta_{0})\quad(\delta_{0} > 0), \quad G(0) = 0 $

和

$ \omega_{1}(d(x)) = \int_{\psi(\omega_{1}(d(x)))}^{\infty} \frac{{\rm d}s}{(k f(s))^{\frac{1}{k}}}, $

可得

$ \lim\limits _{d(x) \rightarrow 0} \frac{G(d(x))}{g(d(x))} = 0, $

$ \lim\limits _{d(x) \rightarrow 0} \frac{G(d(x))g^{\prime}(d(x))}{g^{2}(d(x))} = 1-G_{g}, $

$ \lim\limits _{d(x) \rightarrow 0} \frac{w_{1}(d(x)) f^{\prime}\left(\psi\left(w_{1}(d(x))\right)\right)}{\left(k f\left(\psi\left(w_{1}(d(x))\right)\right)\right)^{(k-1) / k}} = C^{+}_{f}, $

$ \lim\limits _{d(x) \rightarrow 0} \prod\limits_{j = 1}^{N-1} (1-d(x)\kappa_{j}(\bar{x})) = 1. $

由(4.9) 式可得

$ m_{k-1}\beta^{k}\xi_{2+}^{k}\left(k G_{g}+\left(C^{+}_{f}-1\right)(k+1)\right)-(b_{2}+\epsilon)(1+\epsilon) = \epsilon $

和

$ M_{k-1}\beta^{k}\xi_{1-}^{k}\left(k G_{g}+\left(C^{+}_{f}-1\right)(k+1)\right)-(b_{1}-\epsilon)(1-\epsilon) = -\epsilon. $

(Ⅰ) 当$ g $非递减时, 基于条件$ (B_{3}) $, 可知存在与$ \varepsilon $有关的充分小的$ \delta_{\varepsilon} \in (0, \min\{1, \delta_{1}/2\}) $使得对于$ \varrho \in (0, \delta_{\varepsilon}) $有

(4.10) $ \begin{equation} b(x) >\left(b_{1}-\varepsilon\right) g^{k+1}(d(x))\geq\left(b_{1}-\varepsilon\right) g^{k+1}(d(x)-\varrho), \quad x \in D_{\varrho}^{-} = \Omega_{2 \delta_{\varepsilon}} / \bar{\Omega}_{\varrho}, \end{equation} $

(4.11) $ \begin{equation} \left(b_{2}+\varepsilon\right) g^{k+1}(d(x)+\varrho)\geq\left(b_{2}+\varepsilon\right) g^{k+1}(d(x))>b(x), \quad x \in D_{\varrho}^{+} = \Omega_{2 \delta_{\varepsilon}-\varrho}, \end{equation} $

(4.12) $ \begin{equation} 1+\epsilon > \prod\limits_{j = 1}^{N-1} (1-d(x)\kappa_{j}(\bar{x}))>1-\epsilon, \quad x \in \Omega_{2 \delta_{\varepsilon}}, \end{equation} $

且对于$ x \in \Omega_{2 \delta_{\varepsilon}} $有

$ \begin{eqnarray*} &&k M_{k} \beta^{k} \xi_{1-}^{k} \frac{G(d(x))}{g(d(x))} \frac{\xi_{1-} w_{1}(d(x))f^{\prime}\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)}{\big(k f\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)\big)^{(k-1) / k}} \\ && +k M_{k-1} \beta^{k} \xi_{1-}^{k}\left(\beta \frac{\xi_{1-} w_{1}(d(x)) f^{\prime}\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)}{\big(k f\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)\big)^{(k-1) / k}}-\left(\beta-1+\frac{G(d(x)) g^{\prime}(d(x))}{g^{2}(d(x))}\right)\right) \\ && -\left(b_{1}-\varepsilon\right)(1-\varepsilon)<0 \end{eqnarray*} $

和

$ \begin{eqnarray*} &&k m_{k} \beta^{k} \xi_{2+}^{k} \frac{G(d(x))}{g(d(x))} \frac{\xi_{2+} w_{1}(d(x)) f^{\prime}\big(\psi\left(\xi_{2+} w_{1}(d(x))\right)\big)}{\big(k f\big(\psi\left(\xi_{2+}w_{1}(d(x))\right)\big)\big)^{(k-1) / k}}\\ &&+k m_{k-1} \beta^{k} \xi_{2+}^{k}\left(\beta \frac{\xi_{2+} w_{1}(d(x)) f^{\prime}\left(\psi\left(\xi_{2+} w_{1}(d(x))\right)\right)}{\big(k f\big(\psi\left(\xi_{2+} w_{1}(d(x))\right)\big)\big)^{(k-1) / k}}-\left(\beta-1+\frac{G(d(x)) g^{\prime}(d(x))}{g^{2}(d(x))}\right)\right)\\ &&-\left(b_{2}+\varepsilon\right)(1+\varepsilon)>0. \end{eqnarray*} $

由(4.8) 式和(4.12) 式, 得

(4.13) $ \begin{equation} \frac{m_{k-1}}{1+\epsilon} < S_{k-1}\left(\frac{\kappa_{1}(\bar{x})}{1-d(x) \kappa_{1}(\bar{x})}, \cdots , \frac{\kappa_{N-1}(\bar{x})}{1-d(x) \kappa_{N-1}(\bar{x})}\right)<\frac{M_{k-1}}{1-\epsilon}. \end{equation} $

令

$ d_{1}(x) = d(x)-\varrho; \quad d_{2}(x) = d(x)+\varrho; $

(4.14) $ \begin{equation} \bar{z}_{\varepsilon} = \psi\left(\xi_{1-} w_{1}\left(d_{1}(x)\right)\right), x \in D_{\varrho}^{-} ; \quad \underline{z}_{\varepsilon} = \psi\left(\xi_{2+} w_{1}\left(d_{2}(x)\right)\right), x \in D_{\varrho}^{+}. \end{equation} $

由(4.13) 式, 引理2.3和引理4.3, 可知对于$ x \in D_{\varrho}^{-} $有

$ \begin{eqnarray*} &&S_{k}\left(\lambda\left(D^{2} \bar{z}_{\epsilon}\right)\right)-b(x) f\left(\bar{z}_{\epsilon}\right) \\ &\leq& S_{k}\left(\lambda\left(D^{2} \bar{z}_{\epsilon}\right)\right)-\left(b_{1}-\epsilon\right) g^{k+1}(d_{1}(x)) f\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big) \\ & = &\Big(-\xi_{1-} \beta g(d_{1}(x)) G^{\beta-1}(d_{1}(x)) \psi^{\prime}\left(\xi_{1-} w_{1}(d_{1}(x))\right)\Big)^{k} S_{k}\left(\frac{\kappa_{1}(\bar{x})}{1-d(x) \kappa_{1}(\bar{x})}, \cdots , \frac{\kappa_{N-1}(\bar{x})}{1-(x) \kappa_{N-1}(\bar{x})}\right) \\ &&+\Big(-\xi_{1-} \beta g(d_{1}(x)) G^{\beta-1}(d_{1}(x)) \psi^{\prime}\left(\xi_{1-} w_{1}(d_{1}(x))\right)\Big)^{k-1} \xi_{1-} \beta G^{\beta-2}(d_{1}(x))g^{2}(d_{1}(x)) \\ &&\times \left(\xi_{1-} \beta G^{\beta}(d_{1}(x)) \psi^{\prime \prime}\left(\xi_{1-} w_{1}(d_{1}(x))\right)+\psi^{\prime}\left(\xi_{1-} w_{1}(d_{1}(x))\right)\left(\beta-1+\frac{G(d_{1}(x)) g^{\prime}(d_{1}(x))}{g^{2}(d_{1}(x))}\right)\right)\\ && \times S_{k-1}\left(\frac{\kappa_{1}(\bar{x})}{1-d(x) \kappa_{1}(\bar{x})}, \cdots , \frac{\kappa_{N-1}(\bar{x})}{1-d(x) \kappa_{N-1}(\bar{x})}\right)-\left(b_{1}-\varepsilon\right) g^{k+1}\left(d_{1}(x)\right) f\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big) \\ &\leq&\frac{g^{k+1}(d_{1}(x))}{1-\epsilon} f\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big)\left(k M_{k} \beta^{k} \xi_{1-}^{k} \frac{G(d_{1}(x))}{g(d_{1}(x))} \frac{\xi_{1-} w_{1}(d_{1}(x)) f^{\prime}\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big)}{\big(k f\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big)\big)^{(k-1) / k}} \right. \\ &&\left.\quad+k M_{k-1} \beta^{k} \xi_{1-}^{k}\left(\beta \frac{\xi_{1-} w_{1}(d_{1}(x)) f^{\prime}\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big)}{\big(k f\big(\psi\left(\xi_{1-} w_{1}(d_{1}(x))\right)\big)\big)^{(k-1) / k}}-\left(\beta-1+\frac{G(d_{1}(x))g^{\prime}(d_{1}(x))}{g^{2}(d_{1}(x))}\right)\right) \right.\\ && \Bigg.-\left(b_{1}-\varepsilon\right)(1-\varepsilon)\Bigg) \leq 0, \end{eqnarray*} $

这表明在$ D_{\varrho}^{-} $中, $ \bar{z}_{\varepsilon} $是$ k $-Hessian问题(1.1) 的上解.

显然, 在$ D_{\varrho}^{-} $中, $ D^{2} \bar{z}_{\varepsilon}>0 $. 结合引理4.2和4.3的证明, 利用(4.2) 式和引理2.3中$ (1) $–$ (3) $, 可得矩阵$ D^{2} \bar{z}_{\varepsilon} $的$ l $阶主子式

$ \begin{eqnarray*} &&\left(-\xi_{1-} \beta G^{\beta-1}\left(d_{1}(x)\right) g\left(d_{1}(x)\right) \psi^{\prime}\left(\xi_{1-} w_{1}\left(d_{1}(x)\right)\right)\right)^{l} \prod\limits_{j = 1}^{l} \frac{\kappa_{j}(\bar{x})}{1-d(x) \kappa_{j}(\bar{x})} \\ & = &\left(\beta \xi_{1-}\right)^{l} G^{l / k}\left(d_{1}(x)\right) g^{l}\left(d_{1}(x)\right)\left(k f\left(\psi\left(\xi_{1-} w_{1}\left(d_{1}(x)\right)\right)\right)\right)^{l / k} \prod\limits_{j = 1}^{l} \frac{\kappa_{j}(\bar{x})}{1-d(x) \kappa_{j}(\bar{x})} > 0, \end{eqnarray*} $

其中, $ l = 1, 2, \cdots , N-1 $. 因此, 在$ D_{\varrho}^{-} $中, 矩阵$ D^{2} \bar{z}_{\varepsilon} $是正定的.

类似地, 可证在$ D_{\varrho}^{+} $中, $ \underline{z}_{\varepsilon} = \psi\left(\xi_{2+} w_{1}\left(d_{2}(x)\right)\right) $是$ k $-Hessian问题(1.1) 的下解且矩阵$ D^{2} \underline{z}_{\varepsilon} $是正定的.

令$ z \in C^{2}(\Omega) $是$ k $-Hessian问题(1.1) 的一个严格凸解且$ C_{0} > 0 $充分大可使得

(4.15) $ \begin{equation} z\leq \bar{z}_{\varepsilon}+C_{0}, \quad d(x) = 2\delta_{\varepsilon}; \quad \underline{z}_{\varepsilon}\leq z+C_{0}, \quad d(x) = 2\delta_{\varepsilon}-\varrho. \end{equation} $

显然, $ \lim\limits _{d_{1}(x) \rightarrow \varrho}\bar{z}_{\varepsilon}(x) = \infty $, $ z\mid_{\partial \Omega} = +\infty > \underline{z}_{\epsilon}\mid_{\partial \Omega} $. 由引理4.1可得

(4.16) $ \begin{equation} z\leq \bar{z}_{\varepsilon}+C_{0}, \quad x \in D_{\varrho}^{-}, \quad \underline{z}_{\varepsilon}\leq z+C_{0}, \quad x \in D_{\varrho}^{+}, \end{equation} $

即

$ 1-\frac{C_{0}}{\psi\left(\xi_{2+} w_{1}\left(d_{2}(x)\right)\right)} \leq \frac{z(x)}{\psi\left(\xi_{2+} w_{1}\left(d_{2}(x)\right)\right)}, \quad x \in D_{\varrho}^{+} $

和

$ \frac{z(x)}{\psi\left(\xi_{1-} w_{1}\left(d_{1}(x)\right)\right)} \leq 1+\frac{C_{0}}{\psi\left(\xi_{1-} w_{1}\left(d_{1}(x)\right)\right)}, \quad x \in D_{\varrho}^{-}. $

对于$ x \in D_{\varrho}^{-} \cap D_{\varrho}^{+} $, 令$ \varrho \rightarrow 0 $, 有

$ 1-\frac{C_{0}}{\psi\left(\xi_{2+} w_{1}\left(d(x)\right)\right)} \leq \frac{z(x)}{\psi\left(\xi_{2+} w_{1}\left(d(x)\right)\right)} $

和

$ \frac{z(x)}{\psi\left(\xi_{1-} w_{1}\left(d(x)\right)\right)} \leq 1+\frac{C_{0}}{\psi\left(\xi_{1-} w_{1}\left(d(x)\right)\right)}, $

则

(4.17) $ \begin{equation} 1 \leq \lim\limits _{d(x) \rightarrow 0} \inf \frac{z(x)}{\psi\left(\xi_{2+} w_{1}\left(d(x)\right)\right)} \quad \mbox{ 和 } \quad \lim\limits _{d(x) \rightarrow 0} \sup \frac{z(x)}{\psi\left(\xi_{1-} w_{1}\left(d(x)\right)\right)} \leq 1, \end{equation} $

进一步, 在(4.17) 式中, 令$ \varepsilon \rightarrow 0 $可得(4.6) 式.

(Ⅱ) 当$ g $非递增时, 令

$ \omega_{2}(d(x)): = (G(d(x))-G(\varrho))^{\beta}, \quad d(x)>\varrho>0. $

由引理2.3和条件$ (B_{3}) $得

(4.18) $ \begin{equation} \lim\limits _{(d(x), \varrho) \rightarrow (0, 0)} \frac{w_{2}(d(x)) f^{\prime}\left(\psi\left(w_{2}(d(x))\right)\right)}{\left(k f\left(\psi\left(w_{2}(d(x))\right)\right)\right)^{(k-1) / k}} = C^{+}_{f}, \end{equation} $

且存在与$ \epsilon $有关的充分小的$ \delta_{\epsilon} \in (0, \min\{1, \delta_{1}/2\}) $使得

(4.19) $ \begin{equation} \left(b_{1}-\varepsilon\right) g^{k+1}(d(x)) \leq b(x)\leq \left(b_{2}+\varepsilon\right) g^{k+1}(d(x)), x \in \Omega_{2 \delta_{\varepsilon}}. \end{equation} $

令$ \varrho \in (0, \delta_{\epsilon}) $,

(4.20) $ \begin{equation} \bar{z}_{\varepsilon} = \psi\left(\xi_{1-} w_{2}\left(d(x)\right)\right), x \in D_{\varrho}^{-}, \end{equation} $

利用引理2.1, 引理4.3, (4.18) 式, (4.19) 式, 可得

$ 0< \frac{G(d(x))-G(\varrho)}{G(d(x))} <1 $

和

$ \lim\limits _{d(x) \rightarrow 0} \frac{G(d(x))g^{\prime}(d(x))}{g^{2}(d(x))} = 1-G_{g}\leq 0, $

对于$ x \in D_{\varrho}^{-} $, 有

$ \begin{eqnarray*} &&S_{k}\left(\lambda\left(D^{2} \bar{z}_{\epsilon}\right)\right)-b(x) f\left(\bar{z}_{\epsilon}\right) \\ &\leq& S_{k}\left(\lambda\left(D^{2} \bar{z}_{\epsilon}\right)\right)-\left(b_{1}-\epsilon\right) g^{k+1}(d(x)) f\left(\psi\left(\xi_{1-} w_{2}(d(x))\right)\right) \\ & = &\Big(-\xi_{1-} \beta g(d(x)) G^{\beta-1}(d(x)) \psi^{\prime}\left(\xi_{1-} w_{2}(d(x))\right)\Big)^{k} S_{k}\left(\frac{\kappa_{1}(\bar{x})}{1-d(x) \kappa_{1}(\bar{x})}, \cdots , \frac{\kappa_{N-1}(\bar{x})}{1-d(x) \kappa_{N-1}(\bar{x})}\right) \\ &&+\Big(-\xi_{1-} \beta g(d(x)) G^{\beta-1}(d(x)) \psi^{\prime}\left(\xi_{1-} w_{2}(d(x))\right)\Big)^{k-1} \xi_{1-} \beta G^{\beta-2}(d(x))g^{2}(d(x)) \\ && \times \left(\xi_{1-} \beta G^{\beta}(d(x)) \psi^{\prime \prime}\left(\xi_{1-} w_{2}(d(x))\right)+\psi^{\prime}\left(\xi_{1-} w_{2}(d(x))\right)\left(\beta-1+\frac{G(d(x)) g^{\prime}(d(x))}{g^{2}(d(x))}\right)\right)\\ && \times S_{k-1}\left(\frac{\kappa_{1}(\bar{x})}{1-d(x) \kappa_{1}(\bar{x})}, \cdots , \frac{\kappa_{N-1}(\bar{x})}{1-d(x) \kappa_{N-1}(\bar{x})}\right)-\left(b_{1}-\varepsilon\right) g^{k+1}\left(d_{1}(x)\right) f\big(\psi\left(\xi_{1-} w_{2}(d(x))\right)\big) \\ &\leq&\frac{g^{k+1}(d(x))}{1-\epsilon} f\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)\left(k M_{k} \beta^{k} \xi_{1-}^{k} \frac{G(d(x))}{g(d(x))} \frac{\xi_{1-} w_{1}(d(x)) f^{\prime}\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)}{\big(k f\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)\big)^{(k-1) / k}} \right. \\ &&+k M_{k-1} \beta^{k} \xi_{1-}^{k}\bigg(\beta \frac{\xi_{1-} w_{1}(d(x)) f^{\prime} \big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)} {\big(k f\big(\psi\left(\xi_{1-} w_{1}(d(x))\right)\big)\big)^{(k-1)/ k}} \\ && -\bigg(\beta-1+\frac{G(d(x))g^{\prime}(d(x))}{g^{2}(d(x))} \frac{G(d(x))-G(\varrho)}{G(d(x))}\bigg)\bigg) -\left(b_{1}-\varepsilon\right)(1-\varepsilon)\Bigg) \leq 0, \end{eqnarray*} $

这表明在$ D_{\varrho}^{-} $中, $ \overline{z}_{\epsilon} $是$ k $-Hessian问题(1.1) 的上解. 类似地, 可证

(4.21) $ \begin{equation} \underline{z}_{\varepsilon} = \psi\left(\xi_{2+} \left(G(d(x))+G(\varrho)\right)^{\beta} \right), x \in D_{\varrho}^{+}, \end{equation} $

在$ D_{\varrho}^{+} $中, $ \underline{z}_{\epsilon} $是$ k $-Hessian问题(1.1) 的下解.

显然, 根据(Ⅰ) 中相同的证明可知(4.6) 式在(Ⅱ) 中依然成立. 证毕.