Pure Mathematics
Vol.08 No.03(2018), Article ID:25136,7 pages
10.12677/PM.2018.83038

Many Radial Solutions of Singular k-Hessian Equations

Huayuan Sun, Meiqiang Feng*

School of Applied Science, Beijing Information Science & Technology University, Beijing

Received: May 7th, 2018; accepted: May 21st, 2018; published: May 29th, 2018

ABSTRACT

We prove that many nontrivial radial solutions exist for the singular k-Hessian problem $\left\{\begin{array}{l}{S}_{k}\left({D}^{2}u\right)=H\left(|x|\right)f\left(-u\right)，\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in B，\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}x\in \partial B.\end{array}$ Here  is the k-Hessian operator, and B is the unit ball in ${R}^{N}\left(N\ge 2\right)$ . The main interest is that the weight function $H\in C\left(B\right)$ is unbounded as $x\to \partial B$ , and many nontrivial radial solutions to the above k-Hessian problem are derived. Our approach to show existence and multiplicity, exploits fixed point index theory.

Keywords:k-Hessian Equation, Singular Weight Function, Many Radial Solutions, Existence, Fixed Point Index Theory

Copyright © 2018 by authors and Hans Publishers Inc.

1. 引言

K-Hession问题源自几何学，流体力学和其他应用学科。例如，当k = N时，k-Hessian问题可以表示Weingarten曲率或者是反射面形状，参见文献 [1] 。近年来，越来越多作者开始研究k-Hessian问题，并取得了很多优秀的成果，详见文献 [2] - [12] 。

k-Hessian方程的一般形式如下：

$\left\{\begin{array}{l}{S}_{k}\left({D}^{2}u\right)=H\left(x\right)f\left(-u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega ,\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega .\end{array}$

${S}_{k}\left({D}^{2}u\right)={P}_{k}\left(\Lambda \right)=\underset{1\le {i}_{1}<\cdots <{i}_{k}\le N}{\sum }{\lambda }_{{i}_{1}}\cdots {\lambda }_{{i}_{k}},$

$\Lambda =\left({\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{N}\right)$ 是Hessian矩阵 ${D}^{2}u$ 的特征值。在文献 [13] [14] 中，Wang证明了 ${P}_{k}\left(\Lambda \right)$ 表示 $\Lambda$ 中的第k个初等对称多项式。

$\left\{\begin{array}{l}{S}_{k}\left({D}^{2}u\right)=H\left(|x|\right)f\left(-u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in B,\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}x\in \partial B.\end{array}$ (1.1)

1) $f\in {C}^{2}\left(\left[0,+\infty \right)\right),f\left(0\right)=0,f\left(s\right)>\text{0}\left(0\le s<+\infty \right)$

2) $-s{f}^{\prime }\left(-s\right)>kf\left(-s\right)\text{}\left(s<0\right)$

3) $-s{f}^{\prime }\left(-s\right)

2. 几个引理

${S}_{k}\left({D}^{2}u\right)={C}_{N-1}^{k-1}{t}^{1-N}\left(\frac{{t}^{N-k}}{k}{\left({u}^{\prime }\right)}^{k}\right),t=|x|,x\in {R}^{N}.$

$\left\{\begin{array}{l}{C}_{N-1}^{k-1}{t}^{1-N}{\left(\frac{{t}^{N-k}}{k}{\left(-{u}^{\prime }\right)}^{k}\right)}^{\prime }=\lambda H\left(t\right)f\left(-u\right),\text{}0 (2.1)

$\left\{\begin{array}{l}{C}_{N-1}^{k-1}{t}^{1-N}{\left(\frac{{t}^{N-k}}{k}{\left(-{v}^{\prime }\right)}^{k}\right)}^{\prime }=\lambda H\left(t\right)f\left(v\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0 (2.2)

(H1) $f\in C\left({R}^{+},{R}^{+}\right)$ ，其中 ${R}^{+}=\left[0,+\infty \right)$

(H2) 非负函数 $H\in C\left(\left[0,1\right)\right)$${\int }_{0}^{1}H\left(t\right)\text{d}t<+\infty$ ，且其在[0,1]任意子区间上不恒为0。

1) 如果v(t)是问题(2.2)的一个解，那么 $u\left(t\right)=-v\left(t\right)$ 也是问题(2.1)在J上的一个解；

2) 如果u(t)是问题(2.1)的一个解，那么 $v\left(t\right)=-u\left(t\right)$ 也是问题(2.2)在J上的一个解。

$‖v‖=\underset{t\in J}{\mathrm{max}}|v\left(t\right)|$ 。如果 $v\in {C}^{2}\left(0,1\right)\cap {C}^{1}\left[0,1\right)$ ，且满足(2.2)式，那么称v是问题(2.2)的解。

$v\left(t\right)={{\int }_{t}^{1}\left({\int }_{0}^{\tau }k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right)fv\left(s\right)\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau$ (2.3)

$\underset{t\in {J}_{\theta }}{\mathrm{min}}v\left(t\right)\ge \theta ‖v‖,$ (2.4)

$K=\left\{v\in E:v\ge 0,\underset{t\in {J}_{\theta }}{\mathrm{min}v\left(t\right)\ge \theta ‖v‖}\right\}.$

${\Omega }_{\rho }=\left\{v\in K:\underset{t\in J}{\mathrm{min}}v\left(t\right)<\gamma \rho \right\}=\left\{v\in E:\gamma ‖v‖\le \underset{t\le {J}_{\theta }}{\mathrm{min}v\left(t\right)<\gamma \rho }\right\}.$

1) ${\Omega }_{\rho }$ 是K中开集；

2) ${K}_{\gamma \rho }\subset {\Omega }_{\rho }\subset {K}_{\rho }$

3) $v\in \partial {\Omega }_{\rho }⇔\underset{t\in {J}_{\theta }}{\mathrm{min}}v\left(t\right)=\gamma \rho$

4) 如果 $v\in \partial {\Omega }_{\rho }$ ，则有 

$\left(Tv\right)\left(t\right)={\int }_{t}^{1}{\left({\int }_{0}^{\tau }k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right)f\left(v\left(s\right)\right)\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau$ (2.5)

1) 如果对 $\forall x\in \partial {D}_{k},‖Ax‖\le ‖x‖$ ，则 ${i}_{k}\left(A,{D}_{k}\right)=1$

2) 如果  使得对 $\forall x\in \partial {D}_{k}$$\forall \lambda >0$$x\ne Ax+\lambda e$ 成立，则 ${i}_{k}\left(A,{D}_{k}\right)=\text{0}$

3) 设U是K中的开集且 $\stackrel{¯}{U}\in {D}_{k}$ 。如果 ${i}_{k}\left(A,{D}_{k}\right)=1$${i}_{k}\left(A,{U}_{k}\right)=\text{0}$ ，则A在 ${D}_{k}\\stackrel{¯}{{U}_{k}}$ 上有一个不动点。如果条件为 ${i}_{k}\left(A,{D}_{k}\right)=0$${i}_{k}\left(A,{U}_{k}\right)=\text{1}$ ，则结论也成立。

3. 主要结论

$d={\int }_{0}^{1}H\left(s\right)\text{d}s\text{ }\text{ },{d}_{\text{*}}={\int }_{\theta }^{1-\theta }H\left(s\right)\text{d}s,\text{ }\text{ }{f}_{\gamma \rho }^{\rho }=\mathrm{min}\left\{\frac{f\left(v\right)}{{\rho }^{k}}:v\in \left[\gamma \rho ,\rho \right]\right\};$

${f}_{0}^{\rho }=\mathrm{max}\left\{\frac{f\left(v\right)}{{\rho }^{k}}:v\in \left[0,\rho \right]\right\},\text{ }\text{ }\text{ }\text{ }{f}^{\rho }=\underset{v\to \alpha }{\mathrm{lim}}\frac{f\left(v\right)}{{v}^{k}}\left(\alpha :=\infty 或{0}^{+}\right);$

$\frac{\text{1}}{l}={\left\{dk{\left({C}_{N-1}^{k-1}\right)}^{-1}\right\}}^{\frac{1}{k}}\frac{k}{2k-N},\text{ }\text{ }\text{ }\text{ }\text{ }\frac{1}{L}={\left\{{d}_{*}k{\left({C}_{N-1}^{k-1}\right)}^{-1}\right\}}^{\frac{1}{k}}\frac{k}{2k-N}\left[1-{\left(1-\theta \right)}^{\frac{2k-N}{k}}\right].$

(H3)存在 ${\rho }_{\text{1}},{\rho }_{\text{2}},{\rho }_{3}\in \left(0,\text{+}\infty \right)$ ，且 ${\rho }_{\text{1}}<\gamma {\rho }_{2},{\rho }_{2}<{\rho }_{3}$ 使得

${f}_{\text{0}}^{{\rho }_{1}}>{l}^{k},{f}_{\gamma {\rho }_{2}}^{{\rho }_{2}}<{L}^{k},{f}_{0}^{{\rho }_{3}}>{l}^{k},$

(H4)存在 ${\rho }_{\text{1}},{\rho }_{\text{2}},{\rho }_{3}\in \left(0,\text{+}\infty \right)$ ，且 ${\rho }_{\text{1}}<{\rho }_{2}<{\rho }_{3}$ 使得

${f}_{\gamma {\rho }_{1}}^{{\rho }_{1}}>{L}^{k},{f}_{0}^{{\rho }_{2}}<{l}^{k},{f}_{\gamma {\rho }_{3}}^{{\rho }_{3}}>{L}^{k},$

1) 问题(2.2)至少有两个正解 ${v}_{1},{v}_{2}$ 满足 ${v}_{1}\in {\Omega }_{{\rho }_{2}}\\stackrel{¯}{{K}_{{\rho }_{1}}},{v}_{2}\in {K}_{{\rho }_{3}}\\stackrel{¯}{{\Omega }_{{\rho }_{2}}}$

2) 问题(1.1)至少有两个非平凡径向解 ${u}_{1},{u}_{2}$ 满足 ${u}_{1}=-{v}_{1},{u}_{2}=-{v}_{2}$

$\begin{array}{c}\left(Tv\right)\left(t\right)={\int }_{t}^{1}{\left({\int }_{0}^{\tau }k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right)f\left(v\left(s\right)\right)\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau \\ \le {\int }_{0}^{1}{\left({\int }_{0}^{1}k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right)f\left(v\left(s\right)\right)\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau <{\int }_{0}^{1}{\left({\int }_{0}^{1}k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right){l}^{k}{\rho }_{1}^{k}\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau \\ ={\left(k{\left({C}_{N-1}^{k-1}\right)}^{-1}\right)}^{\frac{1}{k}}l{\rho }_{1}{{\int }_{0}^{1}\tau }^{\frac{k-N}{k}}\text{d}\tau {\left({\int }_{0}^{1}{s}^{N-1}H\left(s\right)\text{d}s\right)}^{\frac{1}{k}}\le {\left(k{\left({C}_{N-1}^{k-1}\right)}^{-1}\right)}^{\frac{1}{k}}l{\rho }_{1}{{\int }_{0}^{1}\tau }^{\frac{k-N}{k}}\text{d}\tau {\left({\int }_{0}^{1}H\left(s\right)\text{d}s\right)}^{\frac{1}{k}}\\ ={\left(dk{\left({C}_{N-1}^{k-1}\right)}^{-1}\right)}^{\frac{1}{k}}l{\rho }_{1}\frac{k}{2k-N},\end{array}$

$e\left(t\right)\equiv 1$ ，则 $e\in \partial {K}_{1}$ 。事实上，有

$v\ne Tv+\lambda e,v\in \partial {\Omega }_{{\rho }_{2}},\lambda >0,$

${v}_{0}=T{v}_{0}+{\lambda }_{0}e.$ (3.1)

$\begin{array}{c}{v}_{0}=T{v}_{0}+{\lambda }_{0}e\ge \gamma ‖T{v}_{0}‖+{\lambda }_{0}e\\ \ge \gamma {\int }_{1-\theta }^{\text{1}}{\left({\int }_{\theta }^{1-\theta }k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right)f\left({v}_{0}\left(s\right)\right)\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau +{\lambda }_{0}\\ >\gamma L{\rho }_{2}{\int }_{1-\theta }^{\text{1}}{\left({\int }_{\theta }^{1-\theta }k{\tau }^{k-N}{s}^{N-1}{\left({C}_{N-1}^{k-1}\right)}^{-1}H\left(s\right)\text{d}s\right)}^{\frac{1}{k}}\text{d}\tau +{\lambda }_{0}\text{ }\\ =\gamma L{\rho }_{2}{\left(k{\left({C}_{N-1}^{k-1}\right)}^{-1}\right)}^{\frac{1}{k}}{\int }_{1-\theta }^{1}{\tau }^{\frac{k-N}{k}}\text{d}\tau {\left({\int }_{\theta }^{1-\theta }{s}^{N-1}H\left(s\right)\text{d}s\right)}^{\frac{1}{k}}+{\lambda }_{0}\end{array}$

$\begin{array}{c}\ge \gamma L{\rho }_{2}{\left(k{\left({C}_{N-1}^{k-1}\right)}^{-1}\right)}^{\frac{1}{k}}{\int }_{1-\theta }^{1}{\tau }^{\frac{k-N}{k}}\text{d}\tau {\left({\int }_{\theta }^{1-\theta }H\left(s\right)\text{d}s\right)}^{\frac{1}{k}}+{\lambda }_{0}\\ =\gamma L{\rho }_{2}{\left(k{\left({C}_{N-1}^{k-1}\right)}^{-1}\right)}^{\frac{1}{k}}\frac{k}{2k-N}\left[1-{\left(1-\theta \right)}^{\frac{2k-N}{k}}\right]+{\lambda }_{0}\\ =\gamma {\rho }_{2}+{\lambda }_{0},\end{array}$

1) 如果存在 ${\left\{{\rho }_{i}\right\}}_{i=1}^{2{m}_{0}}\subset \left(0,\text{+}\infty \right),$ 满足 ${\rho }_{\text{1}}<\gamma {\rho }_{2}<{\rho }_{2}<{\rho }_{3}<\gamma {\rho }_{4}<\cdots <{\rho }_{2{m}_{0}}$ ，使得

${f}_{0}^{{\rho }_{2m-1}}<{l}^{k},{f}_{\gamma {\rho }_{2m}}^{{\rho }_{2m}}>{L}^{k},m=1,2,\cdots ,{m}_{0},$

i) 问题(2.2)在K中至少有2m0个正解，

ii) 问题(1.1)至少有2m0个非负径向解。

2) 如果存在 ${\left\{{\rho }_{i}\right\}}_{i=1}^{2{m}_{0}}\subset \left(0,\infty \right),$ 满足 ${\rho }_{\text{1}}<{\rho }_{\text{2}}$${\rho }_{2}<\gamma {\rho }_{3}<{\rho }_{3}<{\rho }_{4}<\gamma {\rho }_{5}<\cdots <{\rho }_{2{m}_{0}+2}$ ，使得

${f}_{\gamma {\rho }_{2m-1}}^{{\rho }_{2m-1}}<{L}^{k},{f}_{0}^{{\rho }_{mk}}>{l}^{k},m=1,2,\cdots ,{m}_{0},$

i) 问题(2.2)在K中至少有2m0－1个正解，

ii) 问题(1.1)至少有2m0－1个非负径向解。

Many Radial Solutions of Singular k-Hessian Equations[J]. 理论数学, 2018, 08(03): 289-295. https://doi.org/10.12677/PM.2018.83038

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