## Boundary Value Problems

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# Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations

Boundary Value Problems20142014:165

DOI: 10.1186/s13661-014-0165-8

Accepted: 25 June 2014

Published: 24 September 2014

## Abstract

The Perron method is used to establish the existence of viscosity multi-valued solutions for a class of Hessian-type equations with prescribed behavior at infinity.

### Keywords

Hessian equation multi-valued solution asymptotic behavior

## 1 Introduction

In [1], [2], the multi-valued solutions of the eikonal equation were studied. Later, in [3], [4] Jin et al. provided a level set method for the computation of multi-valued geometric solutions to general quasilinear partial differential equations and multi-valued physical observables to the semiclassical limit of the Schrödinger equations. In [5], Caffarelli and Li investigated the multi-valued solutions of the Monge-Ampère equation where they first introduced the geometric situation of the multi-valued solutions and obtained the existence, regularity and the asymptotic behavior at infinity of the multi-valued viscosity solutions. In [6] Ferrer et al. used complex variable methods to study the multi-valued solutions for the Dirichlet problems of Monge-Ampère equations on exterior planar domains. Recently, Bao and Dai discussed the multi-valued solutions of Hessian equations, see [7], [8]. Motivated by the above works, in this paper we study the viscosity multi-valued solutions of the Hessian equation
$F\left(\lambda \left({D}^{2}u\right)\right)=\sigma >0,$
(1.1)
where $\sigma$ is a constant and $\lambda \left({D}^{2}u\right)=\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}\right)$ are eigenvalues of the Hessian matrix ${D}^{2}u$. $F$ is assumed to be defined in the symmetric open convex cone $\mathrm{\Gamma }$, with vertex at the origin, containing
satisfies the fundamental structure conditions
(1.2)
and $F$ is a continuous concave function. In addition, $F$ will be assumed to satisfy some more technical assumptions such as
(1.3)
and for any $r\ge 1$, $R>0$,
$F\left(R\left(\frac{1}{{r}^{n-1}},r,\dots ,r\right)\right)\ge F\left(R\left(1,1,\dots ,1\right)\right).$
(1.4)
For every $C>0$ and every compact set $K$ in $\mathrm{\Gamma }$, there is $\mathrm{\Lambda }=\mathrm{\Lambda }\left(C,K\right)$ such that
(1.5)
There exists a number $\mathrm{\Lambda }$ sufficiently large such that at every point $x\in \partial \mathrm{\Omega }$, if ${x}_{1},\dots ,{x}_{n-1}$ represent the principal curvatures of $\partial \mathrm{\Omega }$, then
$\left({x}_{1},\dots ,{x}_{n-1},\mathrm{\Lambda }\right)\in \mathrm{\Gamma }.$
(1.6)

Inequality (1.4) is satisfied by each $k$th root of an elementary symmetric function ($1\le k\le n$) and the $\left(k-l\right)$th root of each quotient of the $k$th elementary symmetric function and the $l$th elementary symmetric function ($1\le l).

## 2 Preliminaries

The geometric situation of the multi-valued function is given in [5]. Let $n\ge 2$, $D\subset {\mathbf{R}}^{n}$ be a bounded domain with smooth boundary $\partial D$, and let $\mathrm{\Sigma }\subset D$ be homeomorphic in ${\mathbf{R}}^{n}$ to an $n-1$ dimensional closed disc. $\partial \mathrm{\Sigma }$ is homeomorphic to an $n-2$ dimensional sphere for $n\ge 3$.

Let $\mathbf{Z}$ be the set of integers and $M=\left(D\setminus \partial \mathrm{\Sigma }\right)×\mathbf{Z}$ denote a covering of $D\setminus \partial \mathrm{\Sigma }$ with the following standard parametrization: fixing ${x}^{\ast }\in D\setminus \partial \mathrm{\Sigma }$ and connecting ${x}^{\ast }$ by a smooth curve in $D\setminus \partial \mathrm{\Sigma }$ to a point $x$ in $D\setminus \partial \mathrm{\Sigma }$. If the curve goes through $\mathrm{\Sigma }$$m\ge 0$ times in the positive direction (fixing such a direction), then we arrive at $\left(x,m\right)$ in $M$. If the curve goes through $\mathrm{\Sigma }$$m\ge 0$ times in the negative direction, then we arrive at $\left(x,-m\right)$ in $M$.

For $k=2,3,\dots$ , we introduce an equivalence relation ‘$\sim k$’ on $M$ as follows: $\left(x,m\right)$ and $\left(y,j\right)$ in $M$ are ‘$\sim k$’ equivalent if $x=y$ and $m-j$ is an integer multiple of $k$. We let ${M}_{k}=M/\sim k$ denote the $k$-sheet cover of $D\setminus \partial \mathrm{\Sigma }$, and let ${\partial }^{\prime }{M}_{k}={\bigcup }_{m=1}^{k}\left(\partial D×\left\{m\right\}\right)$.

We define a distance in ${M}_{k}$ as follows: for any $\left(x,m\right),\left(y,j\right)\in {M}_{k}$, let $l\left(\left(x,m\right),\left(y,j\right)\right)$ denote a smooth curve in ${M}_{k}$ which connects $\left(x,m\right)$ and $\left(y,j\right)$, and let $|l\left(\left(x,m\right),\left(y,j\right)\right)|$ denote its length. Define
$d\left(\left(x,m\right),\left(y,j\right)\right)=\underset{l}{inf}|l\left(\left(x,m\right),\left(y,j\right)\right)|,$

where the infimum is taken over all smooth curves connecting $\left(x,m\right)$ and $\left(y,j\right)$. Then $d\left(\left(x,m\right),\left(y,j\right)\right)$ is a distance.

### Definition 2.1

We say that a function $u$ is continuous at $\left(x,m\right)$ in ${M}_{k}$ if
$\underset{d\left(\left(x,m\right),\left(y,j\right)\right)\to 0}{lim}u\left(y,j\right)=u\left(x,m\right),$

and $u\in {C}^{0}\left({M}_{k}\right)$ if for any $\left(x,m\right)$, $u$ is continuous at $\left(x,m\right)$.

Similarly, we can define $u\in {C}^{\alpha }\left({M}_{k}\right)$, ${C}^{0,1}\left({M}_{k}\right)$ and ${C}^{2}\left({M}_{k}\right)$.

### Definition 2.2

A function $u\in {C}^{2}\left({M}_{k}\right)$ is called admissible if $\lambda \in \overline{\mathrm{\Gamma }}$, where $\lambda =\lambda \left({D}^{2}u\left(x,m\right)\right)=\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}\right)$ are the eigenvalues of the Hessian matrix ${D}^{2}u\left(x,m\right)$.

### Definition 2.3

A function $u\in {C}^{0}\left({M}_{k}\right)$ is called a viscosity subsolution (resp. supersolution) to (1.1) if for any $\left(y,m\right)\in {M}_{k}$ and $\xi \in {C}^{2}\left({M}_{k}\right)$ satisfying
$u\left(x,m\right)\le \left(\text{resp.}\ge \right)\phantom{\rule{0.25em}{0ex}}\xi \left(x,m\right),\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}u\left(y,m\right)=\xi \left(y,m\right),$
we have
$F\left(\lambda \left({D}^{2}\xi \left(y,m\right)\right)\right)\ge \left(\text{resp.}\le \right)\phantom{\rule{0.25em}{0ex}}\sigma .$

### Definition 2.4

A function $u\in {C}^{0}\left({M}_{k}\right)$ is called a viscosity solution to (1.1) if it is both a viscosity subsolution and a viscosity supersolution to (1.1).

### Definition 2.5

A function $u\in {C}^{0}\left({M}_{k}\right)$ is called admissible if for any $\left(y,m\right)\in {M}_{k}$ and any function $\xi \in {C}^{2}\left({M}_{k}\right)$ satisfying $u\left(x,m\right)\le \left(\ge \right)\phantom{\rule{0.25em}{0ex}}\xi \left(x,m\right)$, $x\in {M}_{k}$, $u\left(y,m\right)=\xi \left(y,m\right)$, we have $\lambda \left({D}^{2}\xi \left(y,m\right)\right)\in F$.

### Remark

It is obvious that if $u$ is a viscosity subsolution, then $u$ is admissible.

### Lemma 2.1

Let$\mathrm{\Omega }$be a bounded strictly convex domain in${\mathbf{R}}^{n}$, $\partial \mathrm{\Omega }\in {C}^{2}$, $\phi \in {C}^{2}\left(\overline{\mathrm{\Omega }}\right)$. Then there exists a constant$C$only dependent on$n$, $\phi$and$\mathrm{\Omega }$such that for any$\xi \in \partial \mathrm{\Omega }$, there exists$\overline{x}\left(\xi \right)\in {\mathbf{R}}^{n}$such that

where${w}_{\xi }\left(x\right)=\phi \left(\xi \right)+\frac{\overline{R}}{2}\left({|x-\overline{x}\left(\xi \right)|}^{2}-{|\xi -\overline{x}\left(\xi \right)|}^{2}\right)$for$x\in {\mathbf{R}}^{n}$and$\overline{R}$is a constant satisfying$F\left(\overline{R},\overline{R},\dots ,\overline{R}\right)=\sigma$.

This is a modification of Lemma 5.1 in [5].

### Lemma 2.2

Let$\mathrm{\Omega }$be a domain in${\mathbf{R}}^{n}$and$f\in {C}^{0}\left({\mathbf{R}}^{n}\right)$be nonnegative. Assume that the admissible functions$v\in {C}^{0}\left(\overline{\mathrm{\Omega }}\right)$, $u\in {C}^{0}\left({\mathbf{R}}^{n}\right)$satisfy, respectively,
$\begin{array}{c}F\left(\lambda \left({D}^{2}v\right)\right)\ge f\left(x\right),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega },\hfill \\ F\left(\lambda \left({D}^{2}u\right)\right)\ge f\left(x\right),\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.\hfill \end{array}$
Moreover,
$\begin{array}{c}u\le v,\phantom{\rule{1em}{0ex}}x\in \overline{\mathrm{\Omega }},\hfill \\ u=v,\phantom{\rule{1em}{0ex}}x\in \partial \mathrm{\Omega }.\hfill \end{array}$
Set
$w\left(x\right)=\left\{\begin{array}{ll}v\left(x\right),& x\in \mathrm{\Omega },\\ u\left(x\right),& x\in {\mathbf{R}}^{n}\setminus \mathrm{\Omega }.\end{array}$
Then $w\in {C}^{0}\left({\mathbf{R}}^{n}\right)$ is an admissible function and satisfies in the viscosity sense
$F\left(\lambda \left({D}^{2}w\left(x\right)\right)\right)\ge f\left(x\right),\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$

### Lemma 2.3

Let$B$be a ball in${\mathbf{R}}^{n}$and let$f\in {C}^{0,\alpha }\left(\overline{B}\right)$be positive. Suppose that$\underline{u}\in {C}^{0}\left(\overline{B}\right)$satisfies in the viscosity sense
$F\left(\lambda \left({D}^{2}u\right)\right)\ge f\left(x\right),\phantom{\rule{1em}{0ex}}x\in B.$
Then the Dirichlet problem
$\begin{array}{c}F\left(\lambda \left({D}^{2}u\right)\right)=f\left(x\right),\phantom{\rule{1em}{0ex}}x\in B,\hfill \\ u=\underline{u}\left(x\right),\phantom{\rule{1em}{0ex}}x\in \partial B\hfill \end{array}$

admits a unique admissible viscosity solution$u\in {C}^{0}\left(\overline{B}\right)$.

We refer to [9] for the proof of Lemmas 2.2 and 2.3.

## 3 Existence of viscosity multi-valued solutions with asymptotic behavior

In this section, we establish the existence of viscosity multi-valued solutions with prescribed asymptotic behavior at infinity of (1.1). Let $\mathrm{\Omega }$ be a bounded strictly convex domain with smooth boundary $\partial \mathrm{\Omega }$. Let $\mathrm{\Sigma }$, diffeomorphic to an $\left(n-1\right)$-disc, be the intersection of $\mathrm{\Omega }$ any hyperplane in ${\mathbf{R}}^{n}$. Let $M=\left({\mathbf{R}}^{n}\setminus \partial \mathrm{\Sigma }\right)×\mathbf{Z}$, ${M}_{k}=M/\sim k$ be covering spaces of ${\mathbf{R}}^{n}\setminus \partial \mathrm{\Sigma }$ as in Section 2. $\mathrm{\Sigma }$ divides $\mathrm{\Omega }$ into two open parts, denoted as ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}$. Fixing ${x}^{\ast }\in {\mathrm{\Omega }}^{-}$, we use the convention that going through $\mathrm{\Sigma }$ from ${\mathrm{\Omega }}^{-}$ to ${\mathrm{\Omega }}^{+}$ denotes the positive direction through $\mathrm{\Sigma }$. Our main result is the following theorem.

### Theorem 3.1

Let$k\ge 3$. Then, for any${C}_{m}\in \mathbf{R}$, there exists an admissible viscosity solution$u\in {C}^{0}\left({M}_{k}\right)$of
$F\left(\lambda \left({D}^{2}u\right)\right)=\sigma ,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k}$
(3.1)
satisfying
$\underset{|x|\to \mathrm{\infty }}{lim sup}{|x|}^{n-2}|u\left(x,m\right)-\left(\frac{\overline{R}}{2}{|x|}^{2}+{C}_{m}\right)|<+\mathrm{\infty },$
(3.2)

where$\overline{R}$is a constant satisfying$F\left(\overline{R},\overline{R},\dots ,\overline{R}\right)=\sigma$.

When
$F\left(\lambda \left({D}^{2}u\right)\right)={\sigma }_{k}\left(\lambda \left({D}^{2}u\right)\right),\phantom{\rule{2em}{0ex}}\mathrm{\Gamma }={\mathrm{\Gamma }}_{k}=\left\{\lambda \in {\mathbf{R}}^{n}:{\sigma }_{j}>0,j=1,2,\dots ,k\right\},$
where the $k$th elementary symmetric function
${\sigma }_{k}\left(\lambda \right)=\sum _{{i}_{1}<\cdots <{i}_{k}}{\lambda }_{{i}_{1}}\cdots {\lambda }_{{i}_{k}}$

for $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)$, in [8] Dai obtained the following result.

### Theorem 3.2

Let$k\ge 3$. Then, for any${C}_{m}\in \mathbf{R}$, there exists a$k$-convex viscosity solution$u\in {C}^{0}\left({M}_{k}\right)$of
${\sigma }_{k}\left(\lambda \left({D}^{2}u\right)\right)=1,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k}$
satisfying
$\underset{|x|\to \mathrm{\infty }}{lim sup}\left({|x|}^{k-2}|u\left(x,m\right)-\left(\frac{{C}_{\ast }}{2}{|x|}^{2}+{C}_{m}\right)|\right)<\mathrm{\infty },$

where${C}_{\ast }={\left(\frac{1}{{C}_{n}^{k}}\right)}^{\frac{1}{k}}$.

### Proof of Theorem 3.1

We divide the proof of Theorem 3.1 into two steps.

Step 1. By [10], there is an admissible solution $\mathrm{\Phi }\in {C}^{\mathrm{\infty }}\left(\overline{\mathrm{\Omega }}\right)$ of the Dirichlet problem:
$\begin{array}{c}F\left(\lambda \left({D}^{2}\mathrm{\Phi }\right)\right)={C}_{0}>\sigma ,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega },\hfill \\ \mathrm{\Phi }=0,\phantom{\rule{1em}{0ex}}x\in \partial \mathrm{\Omega }.\hfill \end{array}$
By the comparison principles in [11], $\mathrm{\Phi }\le 0$ in $\mathrm{\Omega }$. Further, by Lemma 2.1, for each $\xi \in \partial \mathrm{\Omega }$, there exists $\overline{x}\left(\xi \right)\in {\mathbf{R}}^{n}$ such that
${W}_{\xi }\left(x\right)<\mathrm{\Phi }\left(x\right),\phantom{\rule{1em}{0ex}}x\in \overline{\mathrm{\Omega }}\setminus \left\{\xi \right\},$
where
${W}_{\xi }\left(x\right)=\frac{\overline{R}}{2}\left({|x-\overline{x}\left(\xi \right)|}^{2}-{|\xi -\overline{x}\left(\xi \right)|}^{2}\right),\phantom{\rule{1em}{0ex}}\xi \in {\mathbf{R}}^{n},$
and ${sup}_{\xi \in \partial \mathrm{\Omega }}|\overline{x}\left(\xi \right)|<\mathrm{\infty }$. Therefore
$\begin{array}{c}{W}_{\xi }\left(\xi \right)=0,\phantom{\rule{2em}{0ex}}{W}_{\xi }\left(x\right)\le \mathrm{\Phi }\left(x\right)\le 0,\phantom{\rule{1em}{0ex}}x\in \overline{\mathrm{\Omega }},\hfill \\ F\left(\lambda \left({D}^{2}{W}_{\xi }\left(x\right)\right)\right)=F\left(\overline{R},\overline{R},\dots ,\overline{R}\right)=\sigma ,\phantom{\rule{1em}{0ex}}\xi \in {\mathbf{R}}^{n}.\hfill \end{array}$
Denote
$W\left(x\right)=\underset{\xi \in \partial \mathrm{\Omega }}{sup}{W}_{\xi }\left(x\right).$
Then
$W\left(x\right)\le \mathrm{\Phi }\left(x\right),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega },$
and by [12]
$F\left(\lambda \left({D}^{2}W\right)\right)\ge \sigma ,\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$
Define
$V\left(x\right)=\left\{\begin{array}{ll}\mathrm{\Phi }\left(x\right),& x\in \mathrm{\Omega },\\ W\left(x\right),& x\in {\mathbf{R}}^{n}\setminus \mathrm{\Omega }.\end{array}$
Then $V\in {C}^{0}\left({\mathbf{R}}^{n}\right)$ is an admissible viscosity solution of
$F\left(\lambda \left({D}^{2}V\right)\right)\ge \sigma ,\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$

Fix some ${R}_{1}>0$ such that $\overline{\mathrm{\Omega }}\subset {B}_{{R}_{1}}\left(0\right)$, where ${B}_{{R}_{1}}\left(0\right)$ is the ball centered at the origin with radius ${R}_{1}$.

Let ${R}_{2}=2{R}_{1}{\overline{R}}^{\frac{1}{2}}$. For $a>1$, defuse
${W}_{a}\left(x\right)=\underset{{B}_{{R}_{1}}}{inf}V+{\int }_{2{R}_{2}}^{|{\overline{R}}^{\frac{1}{2}}x|}{\left({s}^{n}+a\right)}^{\frac{1}{n}}\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$
Then
${D}_{ij}{W}_{a}={\left({|y|}^{n}+a\right)}^{\frac{1}{n}-1}\left[\left({|y|}^{n-1}+\frac{a}{|y|}\right)\overline{R}{\delta }_{ij}-\frac{a{\overline{R}}^{2}{x}_{i}{x}_{j}}{{|y|}^{3}}\right],\phantom{\rule{1em}{0ex}}|x|>0,$
where $y={\overline{R}}^{\frac{1}{2}}x$. By rotating the coordinates, we may set $x=\left(r,0,\dots ,0\right)$. Therefore
${D}^{2}{W}_{a}={\left({R}^{n}+a\right)}^{\frac{1}{n}-1}\overline{R}diag\left({R}^{n-1},{R}^{n-1}+\frac{a}{R},\dots ,{R}^{n-1}+\frac{a}{R}\right),$
where $R=|y|$. Consequently, $\lambda \left({D}^{2}{W}_{a}\right)\in \mathrm{\Gamma }$ for $|x|>0$ and by (1.4)
$F\left(\lambda \left({D}^{2}{W}_{a}\right)\right)\ge F\left(\overline{R},\overline{R},\dots ,\overline{R}\right)=\sigma ,\phantom{\rule{1em}{0ex}}|x|>0.$
Moreover,
${W}_{a}\left(x\right)\le V\left(x\right),\phantom{\rule{1em}{0ex}}|x|\le {R}_{1}.$
(3.3)
Fix some ${R}_{3}>3{R}_{2}$ satisfying
${R}_{3}{\overline{R}}^{\frac{1}{2}}>3{R}_{2}.$
We choose ${a}_{1}>1$ such that for $a\ge {a}_{1}$,
${W}_{a}\left(x\right)>\underset{{B}_{{R}_{1}}}{inf}V+{\int }_{2{R}_{2}}^{3{R}_{2}}{\left({s}^{n}+a\right)}^{\frac{1}{n}}\phantom{\rule{0.2em}{0ex}}ds\ge V\left(x\right),\phantom{\rule{1em}{0ex}}|x|={R}_{3}.$
Then by (3.3) ${R}_{3}\ge {R}_{1}$. According to the definition of ${W}_{a}$,
$\begin{array}{rcl}{W}_{a}\left(x\right)& =& \underset{{B}_{{R}_{1}}}{inf}V+{\int }_{2{R}_{2}}^{|{\overline{R}}^{\frac{1}{2}}x|}s\left({\left(1+\frac{a}{{s}^{n}}\right)}^{\frac{1}{n}}-1\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{2{R}_{2}}^{|{\overline{R}}^{\frac{1}{2}}x|}s\phantom{\rule{0.2em}{0ex}}ds\\ =& \frac{\overline{R}}{2}{|x|}^{2}+{C}_{m}+\underset{{B}_{{R}_{1}}}{inf}V+{\int }_{2{R}_{2}}^{+\mathrm{\infty }}s\left({\left(1+\frac{a}{{s}^{n}}\right)}^{\frac{1}{n}}-1\right)\phantom{\rule{0.2em}{0ex}}ds-{C}_{m}\\ -2{R}_{2}^{2}-{\int }_{|{\overline{R}}^{\frac{1}{2}}x|}^{+\mathrm{\infty }}s\left({\left(1+\frac{a}{{s}^{n}}\right)}^{\frac{1}{n}}-1\right)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.\end{array}$
Let
$\mu \left(m,a\right)=\underset{{B}_{{R}_{1}}}{inf}V+{\int }_{2{R}_{2}}^{+\mathrm{\infty }}s\left({\left(1+\frac{a}{{s}^{n}}\right)}^{\frac{1}{n}}-1\right)\phantom{\rule{0.2em}{0ex}}ds-{C}_{m}-2{R}_{2}^{2}.$
Then $\mu \left(m,a\right)$ is continuous and monotonic increasing for $a$ and when $a\to \mathrm{\infty }$, $\mu \left(m,a\right)\to \mathrm{\infty }$, $1\le m\le k$. Moreover,
(3.4)
Define, for $a\ge {a}_{1}$ and $1\le m\le k$,
${\underline{u}}_{m,a}\left(x\right)=\left\{\begin{array}{ll}max\left\{V\left(x\right),{W}_{a}\left(x\right)\right\}-\mu \left(m,a\right),& |x|\le {R}_{3},\\ {W}_{a}-\mu \left(m,a\right),& |x|\ge {R}_{3}.\end{array}$
Then by (3.4), for $1\le m\le k$,
and by the definition of $V$,
${\underline{u}}_{m,a}\left(x\right)=-\mu \left(m,a\right),\phantom{\rule{1em}{0ex}}x\in \partial \mathrm{\Sigma }.$
Choose ${a}_{2}\ge {a}_{1}$ large enough such that when $a\ge {a}_{2}$,
$\begin{array}{rcl}V\left(x\right)-\mu \left(m,a\right)& =& V\left(x\right)-\underset{{B}_{{R}_{1}}}{inf}V-{\int }_{2{R}_{2}}^{+\mathrm{\infty }}s\left({\left(1+\frac{a}{{s}^{n}}\right)}^{\frac{1}{n}}-1\right)\phantom{\rule{0.2em}{0ex}}ds+{C}_{m}+2{R}_{2}^{2}\\ \le & {C}_{m}\\ \le & \frac{\overline{R}}{2}{|x|}^{2}+{C}_{m},\phantom{\rule{1em}{0ex}}|x|\le {R}_{3}.\end{array}$
Therefore
${\underline{u}}_{m,a}\left(x\right)\le \frac{\overline{R}}{2}{|x|}^{2}+{C}_{m},\phantom{\rule{1em}{0ex}}a\ge {a}_{2},x\in {\mathbf{R}}^{n}.$
By Lemma 2.2, ${\underline{u}}_{m,a}\in {C}^{0}\left({\mathbf{R}}^{n}\right)$ is admissible and satisfies in the viscosity sense
$F\left(\lambda \left({D}^{2}{\underline{u}}_{m,a}\right)\right)\ge \sigma ,\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$
It is easy to see that there exists a continuous function ${a}^{\left(m\right)}\left(a\right)$ such that ${lim}_{a\to \mathrm{\infty }}{a}^{\left(m\right)}\left(a\right)=\mathrm{\infty }$ and $\mu \left(m,{a}^{\left(m\right)}\left(a\right)\right)=\mu \left(1,a\right)$ for $2\le m\le k$. So there exists ${a}_{3}\ge {a}_{2}$ such that ${a}^{\left(m\right)}\left(a\right)>{a}_{2}$ whenever $a\ge {a}_{3}$ and $2\le m\le k$. Let ${a}^{\left(1\right)}\left(a\right)=a$ and define
${\underline{u}}_{a}\left(x,m\right)={\underline{u}}_{m,{a}^{\left(m\right)}\left(a\right)}\left(x\right),\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k}.$
Then, by the definition of ${\underline{u}}_{m,a}$, when $a\ge {a}_{3}$, ${\underline{u}}_{a}\in {C}^{0}\left({M}_{k}\right)$ is a locally admissible function satisfying
and in the viscosity sense
$F\left(\lambda \left({D}^{2}{\underline{u}}_{a}\right)\right)\ge \sigma ,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k}.$

Step 2. We define the solution of (3.1) by the Perron method.

For $a\ge {a}_{3}$, let ${S}_{a}$ denote the set of admissible functions $V\in {C}^{0}\left({M}_{k}\right)$ which can be extended to $\partial \mathrm{\Sigma }$ and satisfies
$\begin{array}{c}F\left(\lambda \left({D}^{2}V\right)\right)\ge \sigma ,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k},\hfill \\ \underset{x\to \overline{x}}{lim}V\left(x,m\right)\le -\mu \left(1,a\right),\phantom{\rule{1em}{0ex}}\overline{x}\in \mathrm{\Gamma },\hfill \\ V\left(x,m\right)\le \frac{\overline{R}}{2}{|x|}^{2}+{C}_{m},\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n},1\le m\le k.\hfill \end{array}$
It is obvious that ${\underline{u}}_{a}\in {S}_{a}$. Hence ${S}_{a}\ne \mathrm{\varnothing }$. Define
${u}_{a}\left(x,m\right)=sup\left\{V\left(x,m\right):V\in {S}_{a}\right\},\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k}.$
Next we prove that ${u}_{a}$ is a viscosity solution of (3.1). From the definition of ${u}_{a}$, it is a viscosity subsolution of (3.1) and satisfies
${u}_{a}\left(x,m\right)\le \frac{\overline{R}}{2}{|x|}^{2}+{C}_{m},\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$

So we need only to prove that ${u}_{a}$ is a viscosity supersolution of (3.1) satisfying (3.2).

For any ${x}_{0}\in {\mathbf{R}}^{n}\setminus \partial \mathrm{\Sigma }$, fix $\epsilon >0$ such that $\overline{B}=\overline{{B}_{\epsilon }\left({x}_{0}\right)}\subset {\mathbf{R}}^{n}\setminus \partial \mathrm{\Sigma }$. Then the lifting of $B$ into ${M}_{k}$ is the $k$ disjoint balls denoted as ${\left\{{B}^{\left(i\right)}\right\}}_{i=1}^{k}$. For any $\left(x,m\right)\in {B}^{\left(i\right)}$, by Lemma 2.3, there exists an admissible viscosity solution $\stackrel{˜}{u}\in {C}^{0}\left(\overline{{B}^{\left(i\right)}}\right)$ to the Dirichlet problem
$\begin{array}{c}F\left(\lambda \left({D}^{2}\stackrel{˜}{u}\right)\right)=\sigma ,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {B}^{\left(i\right)},\hfill \\ \stackrel{˜}{u}={u}_{a},\phantom{\rule{1em}{0ex}}\left(x,m\right)\in \partial {B}^{\left(i\right)}.\hfill \end{array}$
By the comparison principle in [11],
${u}_{a}\le \stackrel{˜}{u},\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {B}^{\left(i\right)}.$
(3.5)
Define
$\psi \left(x,m\right)=\left\{\begin{array}{ll}\stackrel{˜}{u}\left(x,m\right),& \left(x,m\right)\in {B}^{\left(i\right)},\\ {u}_{a}\left(x,m\right),& \left(x,m\right)\in {M}_{k}\setminus {\left\{{B}^{\left(i\right)}\right\}}_{i=1}^{k}.\end{array}$
By Lemma 2.2,
$F\left(\lambda \left({D}^{2}\psi \left(x,m\right)\right)\right)\ge \sigma ,\phantom{\rule{1em}{0ex}}x\in {\mathbf{R}}^{n}.$
As
$\begin{array}{c}F\left(\lambda \left({D}^{2}\stackrel{˜}{u}\right)\right)=\sigma =F\left(\lambda \left({D}^{2}g\right)\right),\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {B}^{\left(i\right)},\hfill \\ \stackrel{˜}{u}={u}_{a}\le g,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in \partial {B}^{\left(i\right)},\hfill \end{array}$
where $g\left(x,m\right)=\frac{\overline{R}}{2}{|x|}^{2}+{C}_{m}$, we have
$\stackrel{˜}{u}\le g,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in \overline{{B}^{\left(i\right)}}$

by the comparison principle in [11]. Therefore $\psi \in {S}_{a}$.

By the definition of ${u}_{a},{u}_{a}\ge \psi$ in ${M}_{k}$. Consequently, $\stackrel{˜}{u}\le {u}_{a}$ in ${B}^{\left(i\right)}$ and further $\stackrel{˜}{u}={u}_{a}$, $\left(x,m\right)\in {B}^{\left(i\right)}$ in view of (3.5). Since ${x}_{0}$ is arbitrary, we conclude that ${u}_{a}$ is an admissible viscosity solution of (3.1).

By the definition of ${u}_{a}$,
${\underline{u}}_{a}\le {u}_{a}\le g,\phantom{\rule{1em}{0ex}}\left(x,m\right)\in {M}_{k},$

so ${u}_{a}$ satisfies (3.2) and we complete the proof of Theorem 3.1. □

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11371110).

## Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology

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