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

## 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.

## 1 Introduction

In , , the multi-valued solutions of the eikonal equation were studied. Later, in ,  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 , 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  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 , . Motivated by the above works, in this paper we study the viscosity multi-valued solutions of the Hessian equation

$F ( λ ( D 2 u ) ) =σ>0,$
(1.1)

where $σ$ is a constant and $λ( D 2 u)=( λ 1 , λ 2 ,…, λ n )$ are eigenvalues of the Hessian matrix $D 2 u$. $F$ is assumed to be defined in the symmetric open convex cone $Γ$, 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 $r≥1$, $R>0$,

$F ( R ( 1 r n − 1 , r , … , r ) ) ≥F ( R ( 1 , 1 , … , 1 ) ) .$
(1.4)

For every $C>0$ and every compact set $K$ in $Γ$, there is $Λ=Λ(C,K)$ such that

(1.5)

There exists a number $Λ$ sufficiently large such that at every point $x∈∂Ω$, if $x 1 ,…, x n − 1$ represent the principal curvatures of $∂Ω$, then

$( x 1 ,…, x n − 1 ,Λ)∈Γ.$
(1.6)

Inequality (1.4) is satisfied by each $k$th root of an elementary symmetric function ($1≤k≤n$) and the $(k−l)$th root of each quotient of the $k$th elementary symmetric function and the $l$th elementary symmetric function ($1≤l).

## 2 Preliminaries

The geometric situation of the multi-valued function is given in . Let $n≥2$, $D⊂ R n$ be a bounded domain with smooth boundary $∂D$, and let $Σ⊂D$ be homeomorphic in $R n$ to an $n−1$ dimensional closed disc. $∂Σ$ is homeomorphic to an $n−2$ dimensional sphere for $n≥3$.

Let $Z$ be the set of integers and $M=(D∖∂Σ)×Z$ denote a covering of $D∖∂Σ$ with the following standard parametrization: fixing $x ∗ ∈D∖∂Σ$ and connecting $x ∗$ by a smooth curve in $D∖∂Σ$ to a point $x$ in $D∖∂Σ$. If the curve goes through $Σ$$m≥0$ times in the positive direction (fixing such a direction), then we arrive at $(x,m)$ in $M$. If the curve goes through $Σ$$m≥0$ times in the negative direction, then we arrive at $(x,−m)$ in $M$.

For $k=2,3,…$ , we introduce an equivalence relation ‘$∼k$’ on $M$ as follows: $(x,m)$ and $(y,j)$ in $M$ are ‘$∼k$’ equivalent if $x=y$ and $m−j$ is an integer multiple of $k$. We let $M k =M/∼k$ denote the $k$-sheet cover of $D∖∂Σ$, and let $∂ ′ M k = ⋃ m = 1 k (∂D×{m})$.

We define a distance in $M k$ as follows: for any $(x,m),(y,j)∈ M k$, let $l((x,m),(y,j))$ denote a smooth curve in $M k$ which connects $(x,m)$ and $(y,j)$, and let $|l((x,m),(y,j))|$ denote its length. Define

$d ( ( x , m ) , ( y , j ) ) = inf l |l ( ( x , m ) , ( y , j ) ) |,$

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

### Definition 2.1

We say that a function $u$ is continuous at $(x,m)$ in $M k$ if

$lim d ( ( x , m ) , ( y , j ) ) → 0 u(y,j)=u(x,m),$

and $u∈ C 0 ( M k )$ if for any $(x,m)$, $u$ is continuous at $(x,m)$.

Similarly, we can define $u∈ C α ( M k )$, $C 0 , 1 ( M k )$ and $C 2 ( M k )$.

### Definition 2.2

A function $u∈ C 2 ( M k )$ is called admissible if $λ∈ Γ ¯$, where $λ=λ( D 2 u(x,m))=( λ 1 , λ 2 ,…, λ n )$ are the eigenvalues of the Hessian matrix $D 2 u(x,m)$.

### Definition 2.3

A function $u∈ C 0 ( M k )$ is called a viscosity subsolution (resp. supersolution) to (1.1) if for any $(y,m)∈ M k$ and $ξ∈ C 2 ( M k )$ satisfying

$u(x,m)≤(resp.≥)ξ(x,m),(x,m)∈ M k andu(y,m)=ξ(y,m),$

we have

$F ( λ ( D 2 ξ ( y , m ) ) ) ≥(resp.≤)σ.$

### Definition 2.4

A function $u∈ C 0 ( M k )$ 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 $u∈ C 0 ( M k )$ is called admissible if for any $(y,m)∈ M k$ and any function $ξ∈ C 2 ( M k )$ satisfying $u(x,m)≤(≥)ξ(x,m)$, $x∈ M k$, $u(y,m)=ξ(y,m)$, we have $λ( D 2 ξ(y,m))∈F$.

### Remark

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

### Lemma 2.1

Let$Ω$be a bounded strictly convex domain in$R n$, $∂Ω∈ C 2$, $φ∈ C 2 ( Ω ¯ )$. Then there exists a constant$C$only dependent on$n$, $φ$and$Ω$such that for any$ξ∈∂Ω$, there exists$x ¯ (ξ)∈ R n$such that

where$w ξ (x)=φ(ξ)+ R ¯ 2 ( | x − x ¯ ( ξ ) | 2 − | ξ − x ¯ ( ξ ) | 2 )$for$x∈ R n$and$R ¯$is a constant satisfying$F( R ¯ , R ¯ ,…, R ¯ )=σ$.

This is a modification of Lemma 5.1 in .

### Lemma 2.2

Let$Ω$be a domain in$R n$and$f∈ C 0 ( R n )$be nonnegative. Assume that the admissible functions$v∈ C 0 ( Ω ¯ )$, $u∈ C 0 ( R n )$satisfy, respectively,

$F ( λ ( D 2 v ) ) ≥ f ( x ) , x ∈ Ω , F ( λ ( D 2 u ) ) ≥ f ( x ) , x ∈ R n .$

Moreover,

$u ≤ v , x ∈ Ω ¯ , u = v , x ∈ ∂ Ω .$

Set

$w(x)= { v ( x ) , x ∈ Ω , u ( x ) , x ∈ R n ∖ Ω .$

Then $w∈ C 0 ( R n )$ is an admissible function and satisfies in the viscosity sense

$F ( λ ( D 2 w ( x ) ) ) ≥f(x),x∈ R n .$

### Lemma 2.3

Let$B$be a ball in$R n$and let$f∈ C 0 , α ( B ¯ )$be positive. Suppose that$u ̲ ∈ C 0 ( B ¯ )$satisfies in the viscosity sense

$F ( λ ( D 2 u ) ) ≥f(x),x∈B.$

Then the Dirichlet problem

$F ( λ ( D 2 u ) ) = f ( x ) , x ∈ B , u = u ̲ ( x ) , x ∈ ∂ B$

admits a unique admissible viscosity solution$u∈ C 0 ( B ¯ )$.

We refer to  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 $Ω$ be a bounded strictly convex domain with smooth boundary $∂Ω$. Let $Σ$, diffeomorphic to an $(n−1)$-disc, be the intersection of $Ω$ any hyperplane in $R n$. Let $M=( R n ∖∂Σ)×Z$, $M k =M/∼k$ be covering spaces of $R n ∖∂Σ$ as in Section 2. $Σ$ divides $Ω$ into two open parts, denoted as $Ω +$ and $Ω −$. Fixing $x ∗ ∈ Ω −$, we use the convention that going through $Σ$ from $Ω −$ to $Ω +$ denotes the positive direction through $Σ$. Our main result is the following theorem.

### Theorem 3.1

Let$k≥3$. Then, for any$C m ∈R$, there exists an admissible viscosity solution$u∈ C 0 ( M k )$of

$F ( λ ( D 2 u ) ) =σ,(x,m)∈ M k$
(3.1)

satisfying

$lim sup | x | → ∞ | x | n − 2 |u(x,m)− ( R ¯ 2 | x | 2 + C m ) |<+∞,$
(3.2)

where$R ¯$is a constant satisfying$F( R ¯ , R ¯ ,…, R ¯ )=σ$.

When

$F ( λ ( D 2 u ) ) = σ k ( λ ( D 2 u ) ) ,Γ= Γ k = { λ ∈ R n : σ j > 0 , j = 1 , 2 , … , k } ,$

where the $k$th elementary symmetric function

$σ k (λ)= ∑ i 1 < ⋯ < i k λ i 1 ⋯ λ i k$

for $λ=( λ 1 ,…, λ n )$, in  Dai obtained the following result.

### Theorem 3.2

Let$k≥3$. Then, for any$C m ∈R$, there exists a$k$-convex viscosity solution$u∈ C 0 ( M k )$of

$σ k ( λ ( D 2 u ) ) =1,(x,m)∈ M k$

satisfying

$lim sup | x | → ∞ ( | x | k − 2 | u ( x , m ) − ( C ∗ 2 | x | 2 + C m ) | ) <∞,$

where$C ∗ = ( 1 C n k ) 1 k$.

### Proof of Theorem 3.1

We divide the proof of Theorem 3.1 into two steps.

Step 1. By , there is an admissible solution $Φ∈ C ∞ ( Ω ¯ )$ of the Dirichlet problem:

$F ( λ ( D 2 Φ ) ) = C 0 > σ , x ∈ Ω , Φ = 0 , x ∈ ∂ Ω .$

By the comparison principles in , $Φ≤0$ in $Ω$. Further, by Lemma 2.1, for each $ξ∈∂Ω$, there exists $x ¯ (ξ)∈ R n$ such that

$W ξ (x)<Φ(x),x∈ Ω ¯ ∖{ξ},$

where

$W ξ (x)= R ¯ 2 ( | x − x ¯ ( ξ ) | 2 − | ξ − x ¯ ( ξ ) | 2 ) ,ξ∈ R n ,$

and $sup ξ ∈ ∂ Ω | x ¯ (ξ)|<∞$. Therefore

$W ξ ( ξ ) = 0 , W ξ ( x ) ≤ Φ ( x ) ≤ 0 , x ∈ Ω ¯ , F ( λ ( D 2 W ξ ( x ) ) ) = F ( R ¯ , R ¯ , … , R ¯ ) = σ , ξ ∈ R n .$

Denote

$W(x)= sup ξ ∈ ∂ Ω W ξ (x).$

Then

$W(x)≤Φ(x),x∈Ω,$

and by 

$F ( λ ( D 2 W ) ) ≥σ,x∈ R n .$

Define

$V(x)= { Φ ( x ) , x ∈ Ω , W ( x ) , x ∈ R n ∖ Ω .$

Then $V∈ C 0 ( R n )$ is an admissible viscosity solution of

$F ( λ ( D 2 V ) ) ≥σ,x∈ R n .$

Fix some $R 1 >0$ such that $Ω ¯ ⊂ B R 1 (0)$, where $B R 1 (0)$ is the ball centered at the origin with radius $R 1$.

Let $R 2 =2 R 1 R ¯ 1 2$. For $a>1$, defuse

$W a (x)= inf B R 1 V+ ∫ 2 R 2 | R ¯ 1 2 x | ( s n + a ) 1 n ds,x∈ R n .$

Then

$D i j W a = ( | y | n + a ) 1 n − 1 [ ( | y | n − 1 + a | y | ) R ¯ δ i j − a R ¯ 2 x i x j | y | 3 ] ,|x|>0,$

where $y= R ¯ 1 2 x$. By rotating the coordinates, we may set $x=(r,0,…,0)$. Therefore

$D 2 W a = ( R n + a ) 1 n − 1 R ¯ diag ( R n − 1 , R n − 1 + a R , … , R n − 1 + a R ) ,$

where $R=|y|$. Consequently, $λ( D 2 W a )∈Γ$ for $|x|>0$ and by (1.4)

$F ( λ ( D 2 W a ) ) ≥F( R ¯ , R ¯ ,…, R ¯ )=σ,|x|>0.$

Moreover,

$W a (x)≤V(x),|x|≤ R 1 .$
(3.3)

Fix some $R 3 >3 R 2$ satisfying

$R 3 R ¯ 1 2 >3 R 2 .$

We choose $a 1 >1$ such that for $a≥ a 1$,

$W a (x)> inf B R 1 V+ ∫ 2 R 2 3 R 2 ( s n + a ) 1 n ds≥V(x),|x|= R 3 .$

Then by (3.3) $R 3 ≥ R 1$. According to the definition of $W a$,

$W a ( x ) = inf B R 1 V + ∫ 2 R 2 | R ¯ 1 2 x | s ( ( 1 + a s n ) 1 n − 1 ) d s + ∫ 2 R 2 | R ¯ 1 2 x | s d s = R ¯ 2 | x | 2 + C m + inf B R 1 V + ∫ 2 R 2 + ∞ s ( ( 1 + a s n ) 1 n − 1 ) d s − C m − 2 R 2 2 − ∫ | R ¯ 1 2 x | + ∞ s ( ( 1 + a s n ) 1 n − 1 ) d s , x ∈ R n .$

Let

$μ(m,a)= inf B R 1 V+ ∫ 2 R 2 + ∞ s ( ( 1 + a s n ) 1 n − 1 ) ds− C m −2 R 2 2 .$

Then $μ(m,a)$ is continuous and monotonic increasing for $a$ and when $a→∞$, $μ(m,a)→∞$, $1≤m≤k$. Moreover,

(3.4)

Define, for $a≥ a 1$ and $1≤m≤k$,

$u ̲ m , a (x)= { max { V ( x ) , W a ( x ) } − μ ( m , a ) , | x | ≤ R 3 , W a − μ ( m , a ) , | x | ≥ R 3 .$

Then by (3.4), for $1≤m≤k$,

and by the definition of $V$,

$u ̲ m , a (x)=−μ(m,a),x∈∂Σ.$

Choose $a 2 ≥ a 1$ large enough such that when $a≥ a 2$,

$V ( x ) − μ ( m , a ) = V ( x ) − inf B R 1 V − ∫ 2 R 2 + ∞ s ( ( 1 + a s n ) 1 n − 1 ) d s + C m + 2 R 2 2 ≤ C m ≤ R ¯ 2 | x | 2 + C m , | x | ≤ R 3 .$

Therefore

$u ̲ m , a (x)≤ R ¯ 2 | x | 2 + C m ,a≥ a 2 ,x∈ R n .$

By Lemma 2.2, $u ̲ m , a ∈ C 0 ( R n )$ is admissible and satisfies in the viscosity sense

$F ( λ ( D 2 u ̲ m , a ) ) ≥σ,x∈ R n .$

It is easy to see that there exists a continuous function $a ( m ) (a)$ such that $lim a → ∞ a ( m ) (a)=∞$ and $μ(m, a ( m ) (a))=μ(1,a)$ for $2≤m≤k$. So there exists $a 3 ≥ a 2$ such that $a ( m ) (a)> a 2$ whenever $a≥ a 3$ and $2≤m≤k$. Let $a ( 1 ) (a)=a$ and define

$u ̲ a (x,m)= u ̲ m , a ( m ) ( a ) (x),(x,m)∈ M k .$

Then, by the definition of $u ̲ m , a$, when $a≥ a 3$, $u ̲ a ∈ C 0 ( M k )$ is a locally admissible function satisfying

and in the viscosity sense

$F ( λ ( D 2 u ̲ a ) ) ≥σ,(x,m)∈ M k .$

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

For $a≥ a 3$, let $S a$ denote the set of admissible functions $V∈ C 0 ( M k )$ which can be extended to $∂Σ$ and satisfies

$F ( λ ( D 2 V ) ) ≥ σ , ( x , m ) ∈ M k , lim x → x ¯ V ( x , m ) ≤ − μ ( 1 , a ) , x ¯ ∈ Γ , V ( x , m ) ≤ R ¯ 2 | x | 2 + C m , x ∈ R n , 1 ≤ m ≤ k .$

It is obvious that $u ̲ a ∈ S a$. Hence $S a ≠∅$. Define

$u a (x,m)=sup { V ( x , m ) : V ∈ S a } ,(x,m)∈ M k .$

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

$u a (x,m)≤ R ¯ 2 | x | 2 + C m ,x∈ R n .$

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

For any $x 0 ∈ R n ∖∂Σ$, fix $ε>0$ such that $B ¯ = B ε ( x 0 ) ¯ ⊂ R n ∖∂Σ$. Then the lifting of $B$ into $M k$ is the $k$ disjoint balls denoted as ${ B ( i ) } i = 1 k$. For any $(x,m)∈ B ( i )$, by Lemma 2.3, there exists an admissible viscosity solution $u ˜ ∈ C 0 ( B ( i ) ¯ )$ to the Dirichlet problem

$F ( λ ( D 2 u ˜ ) ) = σ , ( x , m ) ∈ B ( i ) , u ˜ = u a , ( x , m ) ∈ ∂ B ( i ) .$

By the comparison principle in ,

$u a ≤ u ˜ ,(x,m)∈ B ( i ) .$
(3.5)

Define

$ψ(x,m)= { u ˜ ( x , m ) , ( x , m ) ∈ B ( i ) , u a ( x , m ) , ( x , m ) ∈ M k ∖ { B ( i ) } i = 1 k .$

By Lemma 2.2,

$F ( λ ( D 2 ψ ( x , m ) ) ) ≥σ,x∈ R n .$

As

$F ( λ ( D 2 u ˜ ) ) = σ = F ( λ ( D 2 g ) ) , ( x , m ) ∈ B ( i ) , u ˜ = u a ≤ g , ( x , m ) ∈ ∂ B ( i ) ,$

where $g(x,m)= R ¯ 2 | x | 2 + C m$, we have

$u ˜ ≤g,(x,m)∈ B ( i ) ¯$

by the comparison principle in . Therefore $ψ∈ S a$.

By the definition of $u a , u a ≥ψ$ in $M k$. Consequently, $u ˜ ≤ u a$ in $B ( i )$ and further $u ˜ = u a$, $(x,m)∈ B ( i )$ 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 $u a$,

$u ̲ a ≤ u a ≤g,(x,m)∈ M k ,$

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

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## Acknowledgements

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

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Correspondence to Yongqiang Fu.

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The authors declare that they have no competing interests.

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All authors contributed to each part of this work equally and read and approved the final version of the manuscript.

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