Large solutions of a class of degenerate equations associated with infinity Laplacian

Cuicui Li; Fang Liu

doi:10.1515/ans-2022-0005

Open Access Published by De Gruyter March 8, 2022

Large solutions of a class of degenerate equations associated with infinity Laplacian

Cuicui Li and Fang Liu

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2022-0005

Abstract

In this article, we investigate the boundary blow-up problem

Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω ,

where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ is the highly degenerate operator related to infinity Laplacian which comes from the absolutely minimizing Lipschitz extension and has a close relationship with the random game named tug-of-war. When the function f satisfies the Keller-Osserman-type condition, we establish the existence of the boundary blow-up viscosity solution. Moreover, for the separable case f ( x , u ) = b ( x ) g ( u ) , we establish the asymptotic estimate of the blow-up solution near the boundary under some regular conditions of the domain. Based on the asymptotic estimate and comparison principle, we obtain the uniqueness of the large viscosity solution. During this procedure, we also study the non-existence of the large solution. For the separable case, we show that the Keller-Osserman-type condition is sufficient and necessary for the existence of the boundary blow-up viscosity solution.

Keywords: infinity Laplacian; blow-up solution; comparison principle; existence; boundary asymptotic estimate

MSC 2010: 35J60; 35J70; 35B40

1 Introduction

In this article, we are interested in the following problem:

(1.1) Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω ,

where

Δ ∞ h u ( x ) ≔ ∣ D u ( x ) ∣ h − 3 ⟨ D 2 u D u , D u ⟩ = ∣ D u ( x ) ∣ h − 3 ∑ i , j = 1 n D i u D j u D i j u , h > 1

denotes the degenerate elliptic operator and Ω ⊂ R n ( n ≥ 2 ) is a bounded domain.

For h = 3 , the operator Δ ∞ h u is the infinity Laplacian operator Δ ∞ u = ∑ i , j = 1 n D i u D j u D i j u , which is closely related to the absolutely minimizing Lipschitz extension (AMLE) that was first introduced by Aronsson [1,2, 3,4] in 1960’s. A function u ∈ C ( Ω ) is said to be an absolutely minimizing Lipschitz function in Ω if and only if for any Ω 0 ⊂ ⊂ Ω and any v ∈ C ( Ω ¯ 0 ) with v = u on ∂ Ω 0 , there holds

Lip ( u , Ω 0 ) ≤ Lip ( v , Ω 0 ) .

And a function u ∈ C ( Ω ) is an absolutely minimizing Lipschitz function in Ω if and only if u is a viscosity solution to the infinity Laplacian equation in Ω ,

(1.2) Δ ∞ u = 0 ,

and the viscosity solution u is also called an infinity harmonic function. It was only in 1993 that Jensen [5] showed the equivalence of the AMLE and viscosity solutions of the homogeneous infinity Laplace equation and the uniqueness of AMLE. Armstrong and Smart [6] gave an easy proof for the uniqueness of infinity harmonic functions. Crandall et al. [7] studied the uniqueness of the viscosity solution of the homogeneous infinity Laplacian equation in an unbounded domain. Crandall et al. [8] proved that an infinity harmonic function u ∈ C ( Ω ) is equivalent to it enjoys the so-called comparison property with cones in Ω . Aronsson et al. [9] showed a systematic treatment of the theory of AMLEs. For the inhomogeneous case

Δ ∞ u = f ( x ) ,

Lu and Wang [10] proved the existence and uniqueness of a viscosity solution of the Dirichlet problem when the inhomogeneous term f does not change its sign. They established the fact that the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously. They also showed the comparison with standard function property for the viscosity solutions to the inhomogeneous equation with constant right-hand side, which extended the result of Crandall et al. [8]. In [11], Bhattacharya and Mohammed established the existence or nonexistence of viscosity solutions to the Dirichlet problem

(1.3) Δ ∞ u = f ( x , u ) , in Ω , u = g , on ∂ Ω ,

for f with the sign and the monotonicity restrictions and g ∈ C ( ∂ Ω ) . In [12], they further removed the sign and the monotonicity restrictions and gave the existence result from the general structure condition on f . For more results of absolute minimizers, one can see Barron et al. [13], Barles and Busca [14], Yu [15] etc.

For h = 1 , the operator Δ ∞ h u is the normalized ∞ -Laplacian operator (also called game ∞ -Laplacian)

Δ ∞ G u = 1 ∣ D u ∣ 2 ∑ i , j = 1 n D i u D j u D i j u ,

which has a “tug-of-war” stochastic game approach first introduced by Peres et al. [16]. Roughly speaking, the game is in a set Ω , a running payoff function f in Ω and a terminal payoff function g ( x ) defined on the boundary of Ω . The game is played by two players who take turns depending on the outcome of a coin toss. A token is initially placed at a point x 0 ∈ Ω . During each turn, the player can move the token to any point in an open ball of size ε around the current position. If the player’s move takes the token to a point x g ∈ ∂ Ω , then the game is over and the players are rewarded or penalized through the payoff functions f and g . For ε → 0 , the continuum value function of this game is shown to solve the following game infinity Laplace equation:

(1.4) Δ ∞ G u = f , in Ω , u = g , on ∂ Ω .

Lu and Wang [17] gave a different proof for the Dirichlet problem (1.4) based on partial differential equation (PDE) methods. They established the well-posedness and the comparison with polar quadratic polynomials property for the normalized infinity Laplace equation. Note that the uniqueness is valid when the inhomogeneous term f ( x ) is positive or negative for equations of Δ ∞ G [17,16] and Δ ∞ [10]. A counter-example in [16] showed that the uniqueness of a viscosity solution of the Dirichlet problem for the inhomogeneous equation is invalid if the inhomogeneous term f ( x ) changes its sign. When f is non-positive or non-negative, it is still open whether the uniqueness holds. Lu and Wang [18] constructed a least and a greatest viscosity solution of equation (1.4) with sign-changing inhomogeneous term f in a bounded domain. Furthermore, they proved that these extremal solutions are absolutely extremal solutions. Lu and Wang [19] studied further the uniqueness for a general degenerate elliptic PDE including both normalized infinity Laplacian Δ ∞ G and non-normalized infinity Laplacian Δ ∞ . For more game theory related to infinity Laplacian, we direct the reader to the papers of Barron et al. [20], Evans [21] and Rossi [22]. For generalized tug-of-war games, one can see [15,23, 24,25].

Since the operator Δ ∞ h has no divergent structure, we understand a function u ∈ C ( Ω ) verifying the equation in the viscosity sense. And due to u ( x ) → ∞ as x → ∂ Ω , the viscosity solution to (1.1) is called large solution or boundary blow-up solution.

The infinity Laplacian has received a lot of attention in recent years, notably due to its many applications in image processing [26,27] and optimal mass transportation problems [28,29]. Note that equations (1.1) involved in Δ ∞ h , besides their wide applications, are not only degenerate, singular for 1 < h < 3 , but also not in divergence form and have no variational structure. They constitute a class of operators with particular properties.

In this article, we consider problem (1.1) involved in the family of the infinity Laplacian Δ ∞ h for all h > 1 . The main purpose of this article is to investigate the comparison principle, uniqueness, existence, and non-existence of viscosity solutions. We also investigate the boundary behavior of the blow-up viscosity solutions. The general function f ( x , u ) on the right-hand satisfies the following conditions:

(f1) f ∈ C ( ( Ω , [ 0 , ∞ ) ) , [ 0 , ∞ ) ) , f ( x , t ) is non-decreasing in t for each x ∈ Ω and f ( x , 0 ) = 0 for all x ∈ Ω while f ( x , t ) > 0 for all ( x , t ) ∈ Ω × ( 0 , ∞ ) ;

(f2) For each x ∈ Ω δ = { x ∈ Ω : d ( x ) < δ } , where δ > 0 and d ( x ) = dist ( x , ∂ Ω ) , the Keller-Osserman-type condition is as follows:

(1.5) ∫ 1 ∞ 1 F ∗ ( t ; x , r ) 1 h + 1 d t < ∞ , t > 0 ,

where r > 0 and

F ∗ ( t ; x , r ) ≔ ∫ 0 t f ∗ ( s ; x , r ) d s , f ∗ ( t ; x , r ) ≔ min { f ( z , t ) : z ∈ B ¯ ( x , r ) ∩ Ω ¯ } ,

for each ( x , r ) ∈ Ω ¯ × ( 0 , ∞ ) .

Our main results are summarized as follows.

Theorem 1.1

Let Ω ⊂ R n be a bounded domain. If f satisfies ( f1 ) and ( f2 ), then problem (1.1) admits a non-negative viscosity solution.

In order to prove Theorem 1.1, we first establish the comparison principle and then combine Perron’s method with compactness arguments. For the comparison result, the method we use is the classical perturbation argument for viscosity solutions. Due to the difficulty of the strong degeneracy we must construct suitable double variable function. As for the existence of blow-up solutions, due to the high degeneracy and singularity of (1.1), we adopt Perron’s method. The key point is to construct suitable barrier functions and calculate carefully.

We also establish the non-existence result for problem (1.1).

Theorem 1.2

Let Ω ⊂ R n be a bounded domain. If f satisfies (f1) and the following condition at some x 0 ∈ ∂ Ω ,

(1.6) ∫ 1 ∞ 1 f ∗ ( t ; x 0 , r ) 1 h d t = ∞ , for some r > 0 ,

where

f ∗ ( t ; x , r ) ≔ max { f ( z , t ) : x ∈ Ω ¯ , z ∈ B ¯ ( x , r ) ∩ Ω ¯ } ,

then the problem (1.1) has no viscosity solution.

Due to the high degeneracy of Δ ∞ h , we prove the non-existence of the viscosity solution of (1.1) by the perturbation method. Note that Theorem 1.2 only gives a sufficient condition for the non-existence of the viscosity solution to (1.1), not a necessary condition. In fact, we can give the necessary and sufficient conditions for the non-existence of the viscosity solution in the separable case f ( x , u ) = b ( x ) g ( u ) . Now we first give the condition for the functions b ( ⋅ ) and g ( ⋅ ) .

(b1) b ∈ C ( Ω ¯ , ( 0 , ∞ ) ) ;

(b2) There exist a function k ∈ Λ and a positive constant b 0 such that

(1.7) lim d ( x ) → 0 b ( x ) k h + 1 ( d ( x ) ) = b 0 ,

where Λ represents the set of all positive, monotonic functions k ∈ C 1 which satisfy

(1.8) lim t → 0 + K ( t ) k ( t ) ′ = β < ∞ , where K ( t ) = ∫ 0 t k ( s ) d s .

(g1) g ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) , g ( 0 ) = 0 , g ( t ) > 0 for t > 0 , and g is a non-decreasing function;

(g2) the Keller-Osserman-type condition,

(1.9) ∫ 1 ∞ 1 G ( t ) h + 1 d t < ∞ , where G ( t ) = ∫ 0 t g ( s ) d s .

Remark 1.1

If b satisfies (b1) and g satisfies (g1) and (g2), it is easy to obtain the existence of the viscosity solution to (1.1) for f ( x , u ) = b ( x ) g ( u ) by Theorem 1.1.

Theorem 1.3

Let Ω ⊂ R n be a bounded domain. Suppose that b satisfies ( b1 ) and g satisfies ( g1 ). If g fails to satisfy ( g2 ), then the problem

(1.10) Δ ∞ h u = b ( x ) g ( u ) , in Ω , u = ∞ , on ∂ Ω ,

has no viscosity solution.

Remark that Theorem 1.3 shows that the Keller-Osserman-type condition is necessary for the existence of the blow-up solution. The non-existence of the viscosity solution of (1.10) is analyzed by the method of arguing by contradiction.

When the domain Ω possesses the additional C 1 regularity, we can establish the following asymptotic estimate near the boundary and then the uniqueness of the blow-up solution of (1.10) follows immediately based on the comparison principle.

Theorem 1.4

Let Ω ⊂ R n be a bounded C 1 domain. Suppose that b satisfies ( b1 ) and ( b2 ), g satisfies ( g1 ) and

lim t → ∞ g ( t ) ( h + 1 ) G ( t ) h / ( h + 1 ) ∫ t ∞ G ( s ) − 1 / ( h + 1 ) d s = α ≥ 1 .

If α + β > 1 , where β is as in (1.8), then the viscosity solution u to (1.10) satisfies

(1.11) lim d ( x ) → 0 u ( x ) ϑ ( K ( d ( x ) ) ) = α + β − 1 α b 0 α − 1 h + 1 ,

where ϑ satisfies

∫ ϑ ( t ) ∞ 1 [ ( h + 1 ) G ( s ) ] 1 h + 1 d s = t .

Furthermore, if g ( t ) t h is non-decreasing on ( 0 , ∞ ) , then problem (1.10) has a unique viscosity solution.

One should note that if the regularity assumption of Theorem 1.4 holds, then the distance function is a solution of Δ ∞ h v = 0 near the boundary. Thus, we can perturb the distance function to analyze the boundary asymptotic behavior of the blow-up solutions based on the comparison principle and Karamata’s regular variation theory. And then, by the comparison principle, the uniqueness result of the viscosity solution follows immediately.

In [30], for the particular case h = 1 and f ( u ) = u q , Juutinen and Rossi showed that the Keller-Osserman-type condition

(1.12) ∫ a ∞ 1 F ( t ) 2 d t < ∞ , where F ( t ) = ∫ 0 t f ( s ) d s , a > 0

is necessary and sufficient for the existence of the viscosity solutions to the boundary blow-up problem

(1.13) Δ ∞ G u = u q , in Ω , u = ∞ , on ∂ Ω .

They also established the following boundary asymptotic behavior of the blow-up solution,

u ( x ) ∼ 2 ( q + 1 ) ( q − 1 ) 2 1 q − 1 d ( x ) − 2 q − 1 , d ( x ) → 0 .

For h = 3 , Mohammed and Mohammed [31,32] studied the large solutions to

(1.14) Δ ∞ u = b ( x ) f ( u ) , in Ω , u = ∞ , on ∂ Ω ,

where b ∈ C ( Ω ¯ ) is nonnegative, f ∈ C [ 0 , ∞ ) ∩ C 1 ( 0 , ∞ ) , f ( 0 ) = 0 , f ( s ) > 0 , s > 0 , and f ( s ) is nondecreasing on [ 0 , ∞ ) . They proved that (1.14) has a non-negative viscosity solution if the following Keller-Osserman-type condition holds:

(1.15) ∫ a ∞ 1 4 F ( t ) 4 d t < ∞ , where F ( t ) = ∫ 0 t f ( s ) d s , a > 0 .

This means that if the weight function b ( x ) is bounded, the presence of b ( x ) is not really important for the existence of the large solutions. In [33], Wang et al. studied the exact boundary blow-up estimates with the first expansion when f is regularly varying at infinity with index p > 3 and the weighted function b ( x ) is controlled on the boundary in some manner. Furthermore, for the case of f ( s ) = s p ( 1 + c g ( s ) ) , with c ∈ R and the function g normalized regularly varying with index − q < 0 , they obtained the second expansion of solutions near the boundary based on Karamata regular variation theory. In [34], under appropriate structure conditions on the inhomogeneous term f , Zhang established the following the boundary estimate of large solutions to problem (1.14)

lim d ( x ) → 0 u ( x ) ψ ( K ( d ( x ) ) ) = 1 ,

where ψ satisfies ∫ ψ ( t ) ∞ 1 4 F ( s ) 4 d s = t and F ( t ) = ∫ 0 t f ( s ) d s . In [35], the boundary behavior of the blow-up solutions to problem (1.14) is studied under different conditions on weight function b ( x ) and the nonlinear term f .

We direct the reader to papers [36,37,38]. for more results about boundary behavior of the blow-up solutions.

The rest of the article will be organized as follows. In Section 2, we give the definition of the viscosity solution of the equation Δ ∞ h u = f ( x , u ) . And by the perturbation method of viscosity solutions, we prove the comparison principle to Δ ∞ h u = f ( x , u ) . In Section 3, by combining Perron’s method and compactness arguments, we establish the existence of viscosity solutions to (1.1). In Section 4, we give the non-existence of viscosity solutions to (1.1) and (1.10). Finally, in Section 5, under some regular assumption of the domain, by the Karamata’s regular variation theory and the comparison principle, we give the characteristic of the boundary blow-up solution near the boundary, and then we obtain the uniqueness of viscosity solutions to (1.10) based on the comparison principle.

2 Comparison principles

In this section, we first introduce the concept of the viscosity solutions to equation

(2.1) Δ ∞ h u = f ( x , u ) , in Ω

and then establish the comparison results by the double variables method based on the viscosity solutions theory. It should be pointed out that the operator Δ ∞ h has no divergence structure and in order to give a reasonable explanation when the gradient vanishes, we use the definition of viscosity solutions based on semi-continuous extension and we refer the reader to [17,39,40]. Note that the singularity is removable when h > 1 . Hence, we can rewrite equation (2.1) as

F h ( D 2 u , D u ) = f ( x , u ) , x ∈ Ω ,

where F h : S × ( R n \ { 0 } ) → R and F h ( M , p ) ≔ ∣ p ∣ h − 3 ( M p ) ⋅ p . Here S denotes the set of n × n real symmetric matrices. Since h > 1 , we have lim p → 0 F h ( M , p ) = 0 for arbitrary M ∈ S . That is, the operator Δ ∞ h is continuous for h > 1 . Hence, we can define the continuous extension of F h as follows:

F ¯ h ( M , p ) ≔ F h ( M , p ) , if p ≠ 0 , 0 , if p = 0 .

Now let us define the viscosity solution of equation (2.1).

Definition 2.1

Let Ω ⊂ R n be a bounded domain. An upper semi-continuous function u : Ω → R is said to be a viscosity subsolution to equation (2.1) if and only if for every x 0 ∈ Ω and φ ∈ C 2 ( Ω ) such that u ( x 0 ) = φ ( x 0 ) and u ( x ) < φ ( x ) for all x ∈ Ω near x 0 and x ≠ x 0 , there holds

F ¯ h ( D 2 φ ( x 0 ) , D φ ( x 0 ) ) ≥ f ( x 0 , φ ( x 0 ) ) .

Similarly, a lower semi-continuous function u : Ω → R is said to be a viscosity supersolution to equation (2.1) if and only if for every x 0 ∈ Ω and φ ∈ C 2 ( Ω ) such that u ( x 0 ) = φ ( x 0 ) and u ( x ) > φ ( x ) for all x ∈ Ω near x 0 and x ≠ x 0 , there holds

F ¯ h ( D 2 φ ( x 0 ) , D φ ( x 0 ) ) ≤ f ( x 0 , φ ( x 0 ) ) .

If a continuous function u ∈ C ( Ω ) is both a viscosity subsolution and a viscosity supersolution of equation (2.1), then we say that u is a viscosity solution of equation (2.1).

We can also use super-jets and sub-jets (see [39]) to give the definition of the viscosity subsolution and the viscosity supersolution, respectively. Now we first recall the definition of super-jets and sub-jets.

The second order super-jet of an upper semi-continuous function u at x 0 ∈ Ω is the set

J 2 , + u ( x 0 ) = { ( D φ ( x 0 ) , D 2 φ ( x 0 ) ) : φ ∈ C 2 ( Ω ) and u − φ has a local maximum at x 0 } ,

and its closure is

J ¯ 2 , + u ( x 0 ) ≔ { ( p , M ) ∈ R n × S : ∃ ( x i , p i , M i ) ∈ Ω × R × S such that ( p i , M i ) ∈ J 2 , + u ( x i ) and ( x i , p i , M i ) → ( x 0 , p , M ) } .

Similarly, the second-order sub-jet of a lower semi-continuous function u at x 0 ∈ Ω is the set

J 2 , − u ( x 0 ) = { ( D φ ( x 0 ) , D 2 φ ( x 0 ) ) : φ ∈ C 2 ( Ω ) and u − φ has a local minimum at x 0 } ,

and its closure is

J ¯ 2 , − u ( x 0 ) ≔ { ( p , M ) ∈ R n × S : ∃ ( x i , p i , M i ) ∈ Ω × R × S such that ( p i , M i ) ∈ J 2 , − u ( x i ) and ( x i , p i , M i ) → ( x 0 , p , M ) } .

Definition 2.2

We say u ∈ C ( Ω ) is a viscosity subsolution to (2.1) if

F ¯ h ( M , p ) ≥ f ( x 0 , u ( x 0 ) ) , ∀ ( p , M ) ∈ J ¯ 2 , + u ( x 0 ) , ∀ x 0 ∈ Ω .

Similarly, we say u ∈ C ( Ω ) is a viscosity supersolution to (2.1) if

F ¯ h ( M , p ) ≤ f ( x 0 , u ( x 0 ) ) , ∀ ( p , M ) ∈ J ¯ 2 , − u ( x 0 ) , ∀ x 0 ∈ Ω .

By the definition of the viscosity subsolution of (2.1), it is easy to check that if u and v are both viscosity subsolutions of (2.1), then max { u , v } is also a viscosity subsolution of (2.1). And a similar result is also valid for the viscosity supersolution.

Now we recall the following maximum principle for infinity harmonic functions which can be deduced from Harnack’s inequality [9,41].

Lemma 2.1

(Maximum principle) If u ∈ C ( Ω ¯ ) satisfies Δ ∞ u ≥ 0 in the viscosity sense, then

sup Ω u = sup ∂ Ω u .

Moreover, the supremum occurs only on the boundary of Ω unless u is a constant.

Since the degeneracy at the points where the gradient vanishes and the explosive boundary condition, the general comparison results stated in [39] cannot be applied to equation (2.1). Now we give the comparison result by the double variable method based on the viscosity solution theory.

Theorem 2.1

(Comparison principle) Let Ω ⊂ R n be a bounded domain. Assume that f ( x , t ) ∈ C ( ( Ω ¯ , ( 0 , ∞ ) ) , ( 0 , ∞ ) ) is non-decreasing in t for each x ∈ Ω . Suppose also that u ∈ C ( Ω ¯ ) and v ∈ C ( Ω ¯ ) satisfy

Δ ∞ h u ≥ f ( x , u ) , x ∈ Ω

and

Δ ∞ h v ≤ f ( x , v ) , x ∈ Ω

in the viscosity sense. If

lim inf x → z ∈ ∂ Ω u ( x ) ≤ lim sup x → z ∈ ∂ Ω v ( x ) ,

then there holds u ≤ v in Ω .

Proof

Define

u ε ≔ u + ε ( u − sup ∂ Ω u ) , ε > 0 .

By Lemma 2.1, we have u ≤ sup ∂ Ω u and u ε ≤ u in Ω . Since u is a viscosity subsolution of (2.1), we have

(2.2) Δ ∞ h u ε ≥ ( 1 + ε ) h f ( x , u ) ≥ ( 1 + ε ) h f ( x , u ε ) ≥ f ( x , u ε )

in the viscosity sense.

Now we claim that u ε ≤ v in Ω . We argue by contradiction. Suppose that u ε > v somewhere in Ω . Set

M = sup Ω ( u ε − v ) = u ε ( x 0 ) − v ( x 0 ) > 0 , x 0 ∈ Ω ¯ .

Using the arguments in [39], we double the variables

w j ( x , y ) = u ε ( x ) − v ( y ) − j 4 ∣ x − y ∣ 4 , ( x , y ) ∈ Ω × Ω , j = 1 , 2 , … ,

and let ( x j , y j ) ∈ Ω ¯ × Ω ¯ such that w j ( x j , y j ) = sup ( x , y ) ∈ Ω × Ω w j ( x , y ) . According to Proposition 3.7 in [39], we have

lim j → ∞ M j = lim j → ∞ ( u ε ( x j ) − v ( y j ) − j ∣ x j − y j ∣ 4 / 4 ) = M

and

lim j → ∞ j ∣ x j − y j ∣ 4 / 4 = 0 .

Furthermore, x j → x 0 , y j → x 0 as j → ∞ . Due to M > 0 ≥ sup ∂ Ω ( u ε − v ) , there is an open set Ω 0 such that x 0 , x j , and y j ∈ Ω 0 ⊆ Ω for j → ∞ .

Let

ψ ( x ) = j ∣ x − y j ∣ 4 / 4 , ϕ ( y ) = − j ∣ x j − y ∣ 4 / 4 .

It is clear that the functions u ε − ψ and v − ϕ have a local maximum at x j and a local minimum at y j , respectively. We consider the two cases: either x j ≠ y j or x j = y j for j large enough.

Case 1: If x j = y j , we have D ψ ( x j ) = 0 and D 2 ψ ( x j ) = 0 . Since u ε is a viscosity subsolution to Δ ∞ h u = ( 1 + ε ) h f ( x , u ) , we have

(2.3) ( 1 + ε ) h f ( x j , ψ ( x j ) ) = ( 1 + ε ) h f ( x j , u ε ( x j ) ) ≤ 0 ,

which contradicts to f > 0 .

Case 2: If x j ≠ y j , applying the maximum principle for semi-continuous functions, there exist symmetric matrices X j , Y j ∈ S such that

( j ∣ x j − y j ∣ 2 ( x j − y j ) , X j ) ∈ J ¯ 2 , + u ε ( x j ) , ( j ∣ x j − y j ∣ 2 ( x j − y j ) , Y j ) ∈ J ¯ 2 , − v ( y j ) ,

and

X j 0 0 − Y j ≤ D 2 θ j ( x j , y j ) + 1 j ( D 2 θ j ( x j , y j ) ) 2 ,

where θ j ( x , y ) ≔ j 4 ∣ x − y ∣ 4 .

For simplicity, we denote p j = j ∣ x j − y j ∣ 2 ( x j − y j ) and z j = x j − y j . Then

X j 0 0 − Y j ≤ j ( ∣ z j ∣ 2 + 2 ∣ z j ∣ 4 ) I − I − I I + 16 z j ⊗ z j − z j ⊗ z j − z j ⊗ z j z j ⊗ z j .

This means that

( X j ξ ) ⋅ ξ ≤ ( Y j ξ ) ⋅ ξ , ∀ ξ ∈ R n .

That is Y j − X j ≥ 0 . Again since u ε is a viscosity subsolution to Δ ∞ h u = ( 1 + ε ) h f ( x , u ) and v is a viscosity supersolution to (2.1), we can conclude that

0 ≤ ∣ p j ∣ h − 3 ⟨ X j p j , p j ⟩ − ( 1 + ε ) h f ( x j , u ε ( x j ) ) ≤ ∣ p j ∣ h − 3 ⟨ Y j p j , p j ⟩ − f ( y j , v ( y j ) ) + f ( y j , v ( y j ) ) − ( 1 + ε ) h f ( x j , u ε ( x j ) ) ≤ f ( y j , v ( y j ) ) − ( 1 + ε ) h f ( x j , u ε ( x j ) ) ,

where we have used Y j − X j ≥ 0 . Letting j → ∞ , we obtain

(2.4) f ( x 0 , v ( x 0 ) ) − ( 1 + ε ) h f ( x 0 , u ε ( x 0 ) ) ≥ 0 .

Since f ( x , t ) is non-decreasing in t and u ε ( x 0 ) > v ( x 0 ) , we have f ( x 0 , u ε ( x 0 ) ) ≥ f ( x 0 , v ( x 0 ) ) , which contradicts to (2.4). Hence, we have proven u ε ≤ v in Ω . Letting ε → 0 , we have u ≤ v in Ω .□

Remark 2.1

The comparison principle still holds for f ( x , t ) < 0 in Ω . In fact, we can take v ε ≔ ( 1 + ε ) v with ε > 0 . It is easy to verify that v ε is a viscosity supersolution of (2.1) and then one can immediately obtain the symmetric result.

Based on Theorem 2.1, we can establish the comparison principle of the boundary blow-up solutions.

Theorem 2.2

(Comparison principle) Let Ω ⊂ R n be a bounded domain. Suppose f ( x , t ) ∈ C ( Ω ¯ , ( 0 , ∞ ) ) and f ( x , t ) t h are both positive, non-decreasing in t for each x ∈ Ω .

Suppose also that u ∈ C ( Ω ¯ ) and v ∈ C ( Ω ¯ ) satisfy

Δ ∞ h u ≥ f ( x , u ) , x ∈ Ω

and

Δ ∞ h v ≤ f ( x , v ) , x ∈ Ω

in the viscosity sense. If

(2.5) lim sup x → ∂ Ω u ( x ) v ( x ) ≤ 1 ,

then we have u ≤ v in Ω .

Proof

By (2.5), for any ε > 0 , there exists δ > 0 such that

(2.6) u ≤ ( 1 + ε ) v , in Ω δ ≔ { x ∈ Ω : d ( x ) < δ } .

Since f ( x , t ) t h is positive, non-decreasing in t for each x ∈ Ω , then we obtain

Δ ∞ h ( ( 1 + ε ) v ) = ( 1 + ε ) h Δ ∞ h v ≤ ( 1 + ε ) h f ( x , v ) ≤ f ( x , ( 1 + ε ) v ) .

That is, ( 1 + ε ) v is a viscosity supersolution of (2.1). Since u ≤ ( 1 + ε ) v on ∂ ( Ω ⧹ Ω ¯ δ ) , by the comparison principle, Theorem 2.1, we have

(2.7) u ≤ ( 1 + ε ) v , in Ω ⧹ Ω ¯ δ .

By (2.6) and (2.7), we have

u ≤ ( 1 + ε ) v , in Ω .

Since ε > 0 is arbitrary, then we have u ≤ v in Ω .□

Remark 2.2

If f ( x , u ) = b ( x ) g ( u ) and g ( t ) t h is non-decreasing on ( 0 , ∞ ) , then the comparison principle is still valid.

3 Existence of boundary large solutions

In this section, we investigate conditions on the non-linearity f in problem (1.1) that would lead to existence of viscosity solutions. First, we find radial solutions to

(3.1) Δ ∞ h u = g ( u ) , in B R ( x 0 ) , u = ∞ , on ∂ B R ( x 0 ) ,

for any x 0 ∈ R n with u ( x 0 ) = a ∈ ( 0 , ∞ ) , and prove the existence of viscosity solutions to

(3.2) Δ ∞ h u = f ( x , u ) , in Ω , u = M , on ∂ Ω ,

where M ≥ 1 . And then we establish the existence of blow-up solutions of (1.1) based on Perron’s method and compactness arguments.

Now we concern the solvability of the boundary blow-up problem (3.1) in a ball B R ( x 0 ) .

In (3.1), we can take the radius R → ∞ , that is, B = R n , in which case the boundary condition means u ( x ) → ∞ as ∣ x ∣ → ∞ . We look for viscosity solutions of the form u ( x ) = ζ ( r ) , where r = ∣ x − x 0 ∣ . By direct calculation, we obtain

D u ( x ) = ζ ′ ( r ) x − x 0 ∣ x − x 0 ∣

and

D 2 u ( x ) = ζ ″ ( r ) ( x − x 0 ) ⊗ ( x − x 0 ) ∣ x − x 0 ∣ 2 + ζ ′ ( r ) 1 ∣ x − x 0 ∣ I − ζ ′ ( r ) ( x − x 0 ) ⊗ ( x − x 0 ) ∣ x − x 0 ∣ 3

for x ≠ x 0 , where ⊗ denotes the tensor product. Therefore, u satisfies

(3.3) Δ ∞ h u = g ( u )

if and only if

∣ ζ ′ ( r ) ∣ h − 1 ζ ″ ( r ) = g ( ζ ) ,

which is a one-dimensional case for (3.3).

Theorem 3.1

Assume g satisfies ( g1 ) and ( g2 ). Then there exists 0 < R ∗ ≤ ∞ such that problem (3.1) admits a viscosity solution u ∈ C ( B R ( x 0 ) ) with 0 < R < R ∗ .

Proof

Step 1: Let us consider the initial value problem

(3.4) ∣ ζ ′ ∣ h − 1 ζ ″ = g ( ζ ) , ζ ′ ( 0 ) = 0 , ζ ( 0 ) = a ,

where a > 0 . Multiplying by ζ ′ ( r ) in (3.4) and integrating on ( 0 , r ) , we obtain

(3.5) 1 h + 1 ∣ ζ ′ ( r ) ∣ h + 1 = ∫ a ζ ( r ) g ( t ) d t ≥ g ( a ) ( ζ ( r ) − a ) .

Since ζ ′ ( 0 ) = 0 and ζ ′ ( r ) > 0 in ( 0 , R ) , there holds

ζ ′ ( r ) ( ζ ( r ) − a ) − 1 h + 1 ≥ g ( a ) 1 h + 1 .

Integrating once again on ( 0 , r ) , we have

(3.6) ζ ( r ) − a ≥ C r h + 1 h , r ∈ ( 0 , R ) ,

where C is a positive constant.

Step 2: We claim that the function u ( x ) ≔ ζ ( ∣ x − x 0 ∣ ) is a viscosity solution of

(3.7) Δ ∞ h u = g ( u ) , in B R ( x 0 ) .

Indeed, we note that u ∈ C 2 ( B R ( x 0 ) ⧹ { x 0 } ) ∩ C 1 ( B R ( x 0 ) ) , which implies that u is a classical solution in B R ( x 0 ) ⧹ { x 0 } . Let φ ∈ C 2 ( Ω ) such that u − φ has a local maximum at x 0 ∈ Ω . Since D φ ( x 0 ) = D u ( x 0 ) , we have

u ( x ) − u ( x 0 ) ≤ φ ( x ) − φ ( x 0 ) = 1 2 ⟨ D 2 φ ( x 0 ) ( x − x 0 ) , x − x 0 ⟩ + o ( ∣ x − x 0 ∣ 2 ) .

Taking x = x 0 + t e → , where e → is a unit vector and t > 0 , from (3.6), we have

C t h + 1 h ≤ ζ ( t ) − a = u ( x 0 + t e → ) − u ( x 0 ) ≤ 1 2 ⟨ D 2 φ ( x 0 ) e → , e → ⟩ t 2 + o ( t 2 ) ,

which is impossible because φ is twice differentiable at the point x 0 . Thus, there is no φ ∈ C 2 ( Ω ) such that u − φ attains its local maximum at x 0 ∈ Ω . Therefore, u is a viscosity subsolution of (3.7). Since ζ ′ ( 0 ) = 0 , we have D u ( x 0 ) = 0 . Let φ ∈ C 2 ( Ω ) such that u − φ has a local minimum at x 0 ∈ Ω , and then we have Δ ∞ h φ ( x 0 ) = 0 ≤ g ( u ( x 0 ) ) . That is, u is a viscosity supersolution of (3.7).

Step 3: Next we show that problem (3.4) admits an appropriate solution ζ ∈ C 2 ( ( 0 , R ) ) ∩ C 1 ( [ 0 , R ) ) such that

ζ ( r ) → ∞ , r → R .

Let 0 < R 0 ≤ ∞ such that [ 0 , R 0 ) is the maximal interval for the existence of the viscosity solution. From (3.5), we have

1 h + 1 ∣ ζ ′ ( r ) ∣ h + 1 = ∫ a ζ ( r ) g ( t ) d t = G ( ζ ( r ) ) − G ( a ) , G ( t ) = ∫ 0 t g ( s ) d s .

Since ζ ′ ( r ) > 0 , then

ζ ′ ( r ) ( ( h + 1 ) ( G ( ζ ( r ) ) − G ( a ) ) ) 1 h + 1 = 1 , 0 < r < R 0 .

Integrating this on ( 0 , r ) , we obtain

(3.8) ∫ a ζ ( r ) 1 ( ( h + 1 ) ( G ( t ) − G ( a ) ) ) 1 h + 1 d t = r , 0 < r < R 0 .

By (3.5), we have

G ( t ) − G ( a ) ≥ g ( a ) ( t − a ) , t ≥ a

and

G ( t ) − G ( a ) ≥ a g ( a ) ≥ G ( a ) − G ( 0 ) = G ( a ) , t ≥ 2 a

such that

G ( t ) − G ( a ) ≥ 1 2 G ( t ) , t ≥ 2 a .

And then we obtain

(3.9) 1 h + 1 h + 1 ∫ 2 a ζ ( r ) 1 G ( t ) 1 h + 1 d t ≤ ∫ 2 a ζ ( r ) 1 ( ( h + 1 ) ( G ( t ) − G ( a ) ) ) 1 h + 1 d t ≤ 1 ( h + 1 ) / 2 h + 1 ∫ 2 a ζ ( r ) 1 G ( t ) 1 h + 1 d t .

If g satisfies (g2), let

ψ ( a ) ≔ ∫ a ∞ 1 ( ( h + 1 ) ( G ( t ) − G ( a ) ) ) 1 h + 1 d t , a > 0 .

We have lim a → ∞ ψ ( a ) = 0 . Making the change of variable s = G ( t ) − G ( a ) , we obtain

(3.10) ψ ( a ) = ∫ 0 ∞ 1 g ( G − 1 ( s + G ( a ) ) ) 1 ( ( h + 1 ) s ) 1 h + 1 d s .

It is easy to check that ψ ∈ C ( ( 0 , ∞ ) ) is decreasing. From (3.10) and the Monotone convergence theorem, we have

lim a → 0 ψ ( a ) = ∫ 0 ∞ 1 g ( G − 1 ( s ) ) 1 ( ( h + 1 ) s ) 1 h + 1 d s

and 0 < ψ ( 0 ) ≤ ∞ . Let R ∗ = ψ ( 0 ) . Given 0 < R < R ∗ , we choose a > 0 such that ψ ( a ) = R , that is,

∫ a ∞ 1 ( ( h + 1 ) ( G ( t ) − G ( a ) ) ) 1 h + 1 d t = R .

And we note that ζ ( R ) = ∞ and ζ ( 0 ) = a . Hence, ζ ( ⋅ ) is the desired solution to problem (3.4).□

Remark 3.1

When g satisfies (g2), problem (3.1) has a viscosity solution in a bounded domain Ω .

Remark 3.2

If g fails to satisfy (g2), from (3.8) and the first inequality in (3.9), we have R 0 = ∞ , for each a > 0 . And then problem (3.1) admits a viscosity solution u ∈ C ( R n ) with u ( x 0 ) = a , but there are no viscosity solutions in a bounded domain Ω .

Now we are in the position to prove the existence of viscosity solutions to the Dirichlet problem (3.2).

Lemma 3.1

Let Ω ⊂ R n be a bounded domain and f satisfy ( f1 ). For each M ≥ 1 , there exists a unique, non-negative viscosity solution u M ∈ C ( Ω ¯ ) to problem (3.2).

Proof

The uniqueness follows immediately by the comparison principle, Theorem 2.1. The proof of the existence relies on Perron’s method applied to viscosity solutions. It is obvious that u ¯ = M is a viscosity supersolution of problem (3.2).

Next we want to construct a viscosity subsolution with appropriate boundary value. Let

v z ( x ) = M − C ∣ x − z ∣ α ,

where α ∈ ( 0 , 1 ) , C ≥ 1 , and z ∈ ∂ Ω . Then there is a sufficiently small δ > 0 , independent of C and z , such that

Δ ∞ h v z ( x ) = ( 1 − α ) ( α C ) h ∣ x − z ∣ α h − h − 1 ≥ f ( x , M ) ≥ f ( x , v z ) , x ∈ B δ ( z ) .

Choose C so large that v z ≤ 0 outside B δ / 2 ( z ) , and let

u ̲ ( x ) = max { 0 , sup z ∈ ∂ Ω v z ( x ) } .

Then u ̲ ( x ) is the desired viscosity subsolution. Hence, using the standard Perron’s method, we can obtain the existence of a non-negative viscosity solution u M to problem (3.2).□

We now prove the existence result of problem (1.1).

Proof of Theorem 1.1

For each positive integer M , by Lemma 3.1, we can obtain a unique viscosity solution u M to the following problem:

Δ ∞ h u = f ( x , u ) , in Ω , u = M , on ∂ Ω .

By the comparison principle, Theorem 2.1, we have 0 ≤ u M ≤ u M + 1 for each M ≥ 1 .

Let 0 < ε < δ / 2 and D ε = Ω ⧹ Ω ¯ ε . Since ∂ D ε ⊆ Ω δ , we note that f satisfies (f2) at every point of ∂ D ε . Thus, given z ∈ ∂ D ε , let 0 < r z ≤ ε such that condition (f2) holds. Then, by Theorem 3.1, the problem

Δ ∞ h v = f ∗ ( v ; z , r z ) , in B ( z , r z ) , u = ∞ , on ∂ B ( z , r z )

has a viscosity solution v z . Since

Δ ∞ h u M = f ( x , u M ) ≥ f ∗ ( u M ; z , r z ) , in B ( z , r z ) ,

by the comparison principle, Theorem 2.1, we have u M ≤ v z in B ( z , r z ) for all M . Let

K z ≔ max v z ( x ) : x ∈ B z , r z 2 .

We have u M ≤ K z in B z , r z 2 for all M ≥ 1 . From the open cover U ≔ B z , r z 2 : z ∈ ∂ D ε of ∂ D ε , we pick a finite subcover B z j , r z j 2 : j = 1 , 2 , … , k . Let

K ≔ max { K z j : j = 1 , 2 , … , k } ,

and then, we have

u M ≤ K , on ∂ D ε .

Since K is a viscosity supersolution of Δ ∞ h u = f ( x , u ) , by Theorem 2.1 again, we have u 1 ≤ u M ≤ K in D ε . And by Lemma 2.9 of [9], { u M } is locally uniformly bounded and equicontinuous. Therefore, the sequence { u M } converges to a viscosity solution u ∈ C ( Ω ) locally uniformly as M → ∞ . On observing that u M ≤ u in Ω for all M , it follows that u ( x ) → ∞ as x → ∂ Ω . Hence, u is a viscosity solution of problem (1.1).□

By Theorem 1.1, we can immediately obtain the existence of the viscosity solution for the separable case f ( x , u ) = b ( x ) g ( u ) .

Corollary 3.1

Let Ω ⊂ R n be a bounded domain. Suppose that b satisfies ( b1 ) and g satisfies ( g1 ) and ( g2 ). Then problem (1.10) admits a viscosity solution u ∈ C ( Ω ) .

Remark 3.3

When f ( x , u ) = u q , q > h > 1 , the existence follows immediately by Theorem 1.1. For more results in this case, one can see [42].

4 Non-existence of boundary large solutions

In this section, we show that non-existence results of problem (1.1) and (1.10). Since f has less information, we give a sufficient condition that the viscosity solution of (1.1) does not exist. When f ( x , u ) = b ( x ) g ( u ) , we obtain a necessary and sufficient condition for the non-existence of viscosity solutions.

Proof of Theorem 1.2

We argue by contradiction. Assume that there is a viscosity solution u ∈ C ( Ω ) . For x 0 ∈ ∂ Ω , there exist r > 0 and ε > 0 such that u > ε > 0 in Ω 0 ≔ B ( x 0 , r ) ∩ Ω . Let

β ( t ) = 1 t ∫ t 2 t f ∗ ( s ; x , r ) d s , t > 0 .

One can verify that β ( ⋅ ) is a non-decreasing C 1 function on ( 0 , ∞ ) with β ( 0 ) = 0 , and β ( t ) > 0 for t > 0 . As a consequence of the monotonicity of f ∗ , we note that

(4.1) f ∗ ( t ) ≤ β ( t ) ≤ f ∗ ( 2 t ) , t > 0 .

By inequality (4.1) and condition (1.6), we have

(4.2) ∫ 1 ∞ 1 β ( t ) 1 h d t = ∞ .

We consider the following smooth increasing function η : [ ε , ∞ ) → [ 0 , ∞ ) ,

η ( t ) ≔ ∫ ε t 1 β ( s ) 1 h d s , t ≥ ε .

Setting

w ( x ) ≔ η ( u ( x ) ) , x ∈ Ω 0 ,

we have Δ ∞ h w ≤ 1 in Ω 0 . Indeed, let φ ∈ C 2 ( Ω 0 ) and suppose w − φ has a local minimum at z ∈ Ω 0 . Then for some small δ > 0 ,

w ( x ) ≥ φ ( x ) , x ∈ B δ ( z ) ⊆ Ω 0 .

Since w ( z ) > 0 , we have φ > 0 in B δ ( z ) when δ is small enough. Since η is increasing, we have

u ( z ) = η − 1 ( φ ( z ) ) , u ( x ) ≥ η − 1 ( φ ( x ) ) ,

for x ∈ B δ ( z ) . Let ψ ≔ η − 1 ( φ ) . We have ψ ∈ C 2 ( B δ ( z ) ) and u − ψ has a local minimum at z . Since u is a viscosity solution of (1.1), we see that

(4.3) Δ ∞ h ψ ( z ) ≤ f ( z , u ( z ) ) .

In B δ ( z ) , by direct calculations, we have

D ψ = ( η − 1 ( φ ) ) ′ D φ , D 2 ψ = ( η − 1 ( φ ) ) ″ D φ ⊗ D φ + ( η − 1 ( φ ) ) ′ D 2 φ , ( η − 1 ( φ ) ) ′ = ( β ( η − 1 ( φ ) ) ) 1 h , ( η − 1 ( φ ) ) ″ = 1 h ( β ( η − 1 ( φ ) ) ) 2 − h h β ′ ( η − 1 ( φ ) ) ,

and

Δ ∞ h ψ = ( β ( η − 1 ( φ ) ) ) 1 h D φ h − 1 1 h ( β ( η − 1 ( φ ) ) ) 2 − h h β ′ ( η − 1 ( φ ) ) D φ ⊗ D φ + ( β ( η − 1 ( φ ) ) ) 1 h D 2 φ = 1 h ( β ( η − 1 ( φ ) ) ) 1 h β ′ ( η − 1 ( φ ) ) ∣ D φ ∣ h + 1 + β ( η − 1 ( φ ) ) Δ ∞ h φ ,

where ⊗ denotes the tensor product. Since β is nonnegative and nondecreasing, we obtain

(4.4) Δ ∞ h ψ ≥ β ( η − 1 ( φ ) ) Δ ∞ h φ , x ∈ B δ ( z ) .

By (4.3) and (4.4), we have

f ( z , u ( z ) ) ≥ Δ ∞ h ψ ( z ) ≥ β ( ψ ( z ) ) Δ ∞ h φ .

Recalling (4.1), we obtain

Δ ∞ h φ ( z ) ≤ f ( z , u ( z ) ) β ( u ( z ) ) ≤ 1 .

Thus, we have

(4.5) Δ ∞ h w ≤ 1 , in Ω 0 .

By (4.2) and u = ∞ on ∂ Ω , we obtain w = ∞ on B ( x 0 , r ) ∩ ∂ Ω .

We now proceed to show that the conclusion in (4.5) leads to a contradiction. Let α be a continuous function supported on ∂ Ω 0 and

(4.6) α ( x ) = 1 , x ∈ B ( x 0 , r / 3 ) ∩ ∂ Ω , 0 , x ∈ ∂ Ω 0 ⧹ ( B ( x 0 , r / 2 ) ∩ ∂ Ω ) .

From [19,43], we obtain the existence and uniqueness of a viscosity solution τ of the following problem:

Δ ∞ h τ = 1 , in Ω 0 , τ = α , on ∂ Ω 0 .

And then for any constant k ≥ 1 , the function τ k ( x ) ≔ k τ ( x ) satisfies

Δ ∞ h τ k = k h ≥ 1 , in Ω 0 , τ k = k α , on ∂ Ω 0

in the viscosity sense. By the comparison principle, Theorem 2.2, there holds τ k ≤ w in Ω 0 for any k ≥ 1 . Due to Δ ∞ h τ = ∣ D τ ∣ h − 3 Δ ∞ τ = 1 > 0 , by Lemma 2.1, there exists x 1 ∈ Ω 0 such that τ ( x 1 ) > 0 . Then we have τ k ( x 1 ) = k τ ( x 1 ) ≤ w ( x 1 ) for all k ≥ 1 , which leads to a contradiction.□

For the separable case f ( x , t ) = b ( x ) g ( t ) , we can also obtain the non-existence result, which is an improvement of Theorem 1.2.

Proof of Theorem 1.3

We argue by contradiction. Let g 0 ( t ) ≔ M g ( t ) and M ≔ max Ω b . If u is a viscosity solution to (1.10) in Ω , then

Δ ∞ h u ≤ g 0 ( u ) .

Since g fails to satisfy (g2), given

a > u ( x 0 ) , x 0 ∈ Ω ,

we apply Remark 3.2 to find a viscosity solution v ∈ C ( R n ) of Δ ∞ h u = g 0 ( u ) in R n with v ( x 0 ) = a . Since Ω is bounded, we see that v ≤ u in Ω , by the comparison principle, Theorem 2.1. In particular,

a = v ( x 0 ) ≤ u ( x 0 ) ,

which is impossible.□

Theorem 1.3 together with Corollary 3.1, provides a necessary and sufficient condition for problem (1.10) to have a viscosity solution.

5 Boundary asymptotic estimates

In this section, we investigate the asymptotic behavior near the boundary of the blow-up solutions to problem (1.10). Our approach relies on Karamata’s regular variation theory.

Next we recall the definitions and properties of the regularly varying functions which we will use in the sequel. See for example [44,45].

Definition 5.1

A positive measurable function f : [ a , ∞ ) → ( 0 , ∞ ) , for some a > 0 , is called regularly varying at infinity with index ρ ∈ R , written f ∈ RV ρ , if for each ξ > 0 and some ρ ∈ R ,

lim s → ∞ f ( ξ s ) f ( s ) = ξ ρ .

In particular, when ρ = 0 , f is called slowly varying at infinity.

Clearly, if f ∈ RV ρ , then L ( s ) ≔ f ( s ) / s ρ is slowly varying at infinity. And a positive measurable function g defined on ( 0 , a ) for some a > 0 is regularly varying at zero with index ρ ∈ R and write g ∈ RV Z ρ , if g ( 1 / s ) ∈ RV − ρ .

Definition 5.2

A positive measurable function f : [ A , ∞ ) → ( 0 , ∞ ) , for some A > 0 , is called rapidly varying at infinity, if for each ρ > 1 , there holds

lim t → ∞ f ( t ) t ρ = ∞ .

Proposition 5.1

(Representation theorem) A function L is slowly varying at infinity if and only if it can be written in the form:

L ( s ) = c ( s ) exp ∫ a 1 s y ( t ) t d t , s ≥ a 1 ,

for some a 1 ≥ a , where the functions c and y are measurable and for s → ∞ , y ( s ) → 0 , and c ( s ) → c 0 , c 0 > 0 .

Particularly, if c ( s ) ≡ c 0 , then

L ˆ ( s ) ≔ c 0 exp ∫ a 1 s y ( t ) t d t , s ≥ a 1 ,

is normalized slowly varying at infinity and

f ( s ) = s ρ L ˆ ( s ) , s ≥ a 1 ,

is normalized regularly varying at infinity with index ρ and write f ∈ NRV ρ . Similarly, g is called normalized regularly varying at zero with index ρ , written g ∈ NRV Z ρ , if g ( 1 / s ) ∈ NRV − ρ .

If g satisfies (g2) and ϑ satisfies

(5.1) ∫ ϑ ( t ) ∞ 1 [ ( h + 1 ) G ( s ) ] 1 h + 1 d s = t ,

then we observe that ϑ is a decreasing function with ϑ ( t ) → ∞ as t → 0 + . Furthermore, a direct computation shows that

(5.2) ∣ ϑ ′ ( t ) ∣ h − 1 ϑ ″ ( t ) = g ( ϑ ( t ) ) , 0 < t < ψ ( 0 ) ,

and

(5.3) − ϑ ′ ( t ) = ( h + 1 ) G ( ϑ ( t ) ) h + 1 .

Now we recall some properties of the normalized regularly varying functions, which can be found in [46].

Lemma 5.1

Let g ∈ C ( [ 0 , ∞ ) ) with g ( 0 ) = 0 and g ( t ) > 0 for t > 0 .

Suppose that the following condition holds:
(5.4) lim t → ∞ g ( t ) ( h + 1 ) G ( t ) h / ( h + 1 ) ∫ t ∞ G ( s ) − 1 / ( h + 1 ) d s = α ≥ 1 .
Then G ∈ NRV α ( h + 1 ) α − 1 and g ∈ NRV α ( h + 1 ) α − 1 − 1 if α > 1 , whereas G is rapidly varying at infinity if α = 1 .
If α > 1 and G ∈ NRV α ( h + 1 ) α − 1 , then g satisfies ( g2 ) and (5.4).
If G is rapidly varying at infinity, then g satisfies ( g2 ) too.
Suppose that the function ϑ is the inverse of the decreasing function (5.1) and the condition (5.4) holds, then
(5.5) − ϑ ′ ∈ NRV Z − α , ϑ ∈ NRV Z 1 − α ,
and
(5.6) lim t → 0 [ ( h + 1 ) G ( ϑ ( t ) ) ] h / ( h + 1 ) t g ( ϑ ( t ) ) = 1 α .

Now, we are ready to prove the boundary estimate for the viscosity solution u ∈ C ( Ω ) of the problem (1.10) based on the Karamata’s regular variation theory.

Proof of Theorem 1.4

Denote Ω δ ≔ { x ∈ Ω : d ( x ) < δ } for some δ > 0 . Since Ω is a C 1 bounded domain, we have d ( x ) ∈ C 1 ( Ω δ ) . Moreover, ∣ D d ( x ) ∣ = 1 in Ω δ . Then it is obvious that Δ ∞ h d ( x ) = 0 in the viscosity sense in Ω δ . For k ∈ Λ on ( 0 , δ ) and any ε ∈ ( 0 , b 0 / 2 ) > 0 , by (1.7), we have

(5.7) b 0 − ε ≤ b ( x ) k h + 1 ( d ( x ) ) ≤ b 0 + ε , d ( x ) < δ .

Now we construct the comparison functions. We divide it into two cases.

Case (i): First we will show that if k ∈ Λ is non-increasing and δ 0 = δ 0 ( ε ) ∈ ( 0 , δ / 2 ) , then

u ∗ ( x ) = ϑ ( c 1 [ K ( d ( x ) ) − K ( δ 0 ) ] ) , x ∈ Ω δ ⧹ Ω ¯ δ 0 ≕ Ω − ,

and

u ∗ ( x ) = ϑ ( c 2 [ K ( d ( x ) ) + K ( δ 0 ) ] ) , x ∈ Ω δ − δ 0 ≕ Ω + ,

are a viscosity supersolution and a viscosity subsolution of (1.10) in Ω − and in Ω + , respectively, where

(5.8) c 1 = α ( b 0 − ε ) α + β − 1 1 h + 1 and c 2 = α ( b 0 + ε ) α + β − 1 1 h + 1

are positive constants.

Now we prove that u ∗ is a viscosity supersolution of (1.10) in Ω − . The proof that u ∗ is a viscosity subsolution is similar and we omit it. For any x 0 ∈ Ω − and φ ∈ C 2 ( Ω − ) , u ∗ − φ attains local minimum at x 0 . We set

w ( t ) ≔ ϑ ( c 1 [ K ( t ) − K ( δ 0 ) ] ) , t ∈ ( δ 0 , δ ) ,

and ξ is the inverse of w . Since w is decreasing, ξ is decreasing in ( δ 0 , δ ) . Let

ψ ≔ ξ ( φ ) ∈ C 2 ( Ω − ) .

It is easy to verify that

(5.9) Δ ∞ h ψ = ∣ ξ ′ ( φ ) ∣ h − 1 ∣ D φ ∣ h + 1 ξ ″ ( φ ) + ∣ ξ ′ ( φ ) ∣ h − 1 ξ ′ ( φ ) Δ ∞ h φ .

Since Δ ∞ h d ( x ) = 0 , we have Δ ∞ h ψ ( x 0 ) ≥ 0 by the definition of viscosity solution. Then by (5.9) and ξ ′ < 0 , we have

Δ ∞ h φ ( x 0 ) ≤ − ( ξ ′ ( φ ( x 0 ) ) ) − 1 ξ ″ ( φ ( x 0 ) ) ∣ D φ ( x 0 ) ∣ h + 1 .

Note that ∣ D d ( x ) ∣ = 1 in Ω − and d − ψ has a local maximum at x 0 , we have

1 = ∣ D d ( x 0 ) ∣ = ∣ ξ ′ ( φ ( x 0 ) ) D φ ( x 0 ) ∣ .

Then,

(5.10) Δ ∞ h φ ( x 0 ) ≤ − ∣ ξ ′ ( φ ( x 0 ) ) ∣ − h − 1 ξ ′ ( φ ( x 0 ) ) ξ ″ ( φ ( x 0 ) ) = ∣ ξ ′ ( φ ( x 0 ) ) ∣ − h − 2 ξ ″ ( φ ( x 0 ) ) .

For convenience, let ϕ ( t ) = c 1 [ K ( ξ ( t ) ) − K ( δ 0 ) ] . A direct calculation shows that

ξ ′ ( t ) = [ c 1 ϑ ′ ( ϕ ( t ) ) k ( ξ ( t ) ) ] − 1 , ξ ″ ( t ) = − [ c 1 ϑ ′ ( ϕ ( t ) ) k ( ξ ( t ) ) ] − 3 [ ϑ ″ ( ϕ ( t ) ) ( c 1 k ( ξ ( t ) ) ) 2 + c 1 k ′ ( ξ ( t ) ) ϑ ′ ( ϕ ( t ) ) ] .

From (5.2), (5.3), and (5.10), we have

(5.11) Δ ∞ h φ ( x 0 ) ≤ ∣ ξ ′ ( φ ( x 0 ) ) ∣ − h − 2 ξ ″ ( φ ( x 0 ) ) = c 1 ∣ ϑ ′ ( ϕ 0 ( x 0 ) ) k ( d ( x 0 ) ) ∣ h − 1 [ ϑ ″ ( ϕ 0 ( x 0 ) ) ( c 1 k ( d ( x 0 ) ) ) 2 + c 1 ϑ ′ ( ϕ 0 ( x 0 ) ) k ′ ( d ( x 0 ) ) ] = k ( d ( x 0 ) ) h + 1 g ( u ∗ ( x 0 ) ) c 1 h + 1 ∣ ϑ ′ ( ϕ 0 ( x 0 ) ) ∣ h − 1 ϑ ″ ( ϕ 0 ( x 0 ) ) g ( ϑ ( ϕ 0 ( x 0 ) ) ) + c 1 h k ′ ( d ( x 0 ) ) ∣ ϑ ′ ( ϕ 0 ( x 0 ) ) ∣ h − 1 ϑ ′ ( ϕ 0 ( x 0 ) ) k 2 ( d ( x 0 ) ) g ( ϑ ( ϕ 0 ( x 0 ) ) ) ≤ k ( d ( x 0 ) ) h + 1 g ( u ∗ ( x 0 ) ) c 1 h + 1 + c 1 h k ′ ( d ( x 0 ) ) K ( d ( x 0 ) ) k 2 ( d ( x 0 ) ) ∣ ϑ ′ ( ϕ 0 ( x 0 ) ) ∣ h − 1 ϑ ′ ( ϕ 0 ( x 0 ) ) K ( d ( x 0 ) ) g ( ϑ ( ϕ 0 ( x 0 ) ) ) ≤ k ( d ( x 0 ) ) h + 1 g ( u ∗ ( x 0 ) ) c 1 h + 1 − c 1 h + 1 k ′ ( d ( x 0 ) ) K ( d ( x 0 ) ) k 2 ( d ( x 0 ) ) [ ( h + 1 ) G ( ϑ ( ϕ 0 ( x 0 ) ) ) ] h h + 1 ϕ 0 ( x 0 ) g ( ϑ ( ϕ 0 ( x 0 ) ) ) ,

where ϕ 0 ( x 0 ) ≔ c 1 [ K ( d ( x 0 ) ) − K ( δ 0 ) ] . And when d ( x 0 ) → 0 , ϑ ( d ( x 0 ) ) → ∞ . For each k ∈ Λ , it is easy to obtain

(5.12) lim t → 0 + k ′ ( t ) K ( t ) k 2 ( t ) = 1 − β .

Then using (5.6) and (5.12), we obtain

c 1 h + 1 − c 1 h + 1 k ′ ( d ( x 0 ) ) K ( d ( x 0 ) ) k 2 ( d ( x 0 ) ) [ ( h + 1 ) G ( ϑ ( ϕ 0 ( x 0 ) ) ) ] h h + 1 ϕ 0 ( x 0 ) g ( ϑ ( ϕ 0 ( x 0 ) ) ) → c 1 h + 1 α + β − 1 α , d ( x 0 ) → 0 .

By (5.7), we have

(5.13) b ( x 0 ) g ( u ∗ ( x 0 ) ) ≥ ( ℓ − ε ) k h + 1 ( d ( x 0 ) ) g ( u ∗ ( x 0 ) ) .

Combining with (5.8), (5.11), and (5.13), we obtain

Δ ∞ h φ ( x 0 ) ≤ b ( x 0 ) g ( u ∗ ( x 0 ) ) .

That is, u ∗ is a viscosity supersolution.

Case (ii): If k ∈ Λ is non-decreasing and δ 0 = δ 0 ( ε ) ∈ ( 0 , δ / 2 ) , we define

u ∗ ( x ) = ϑ ( c 1 K ( d ( x ) − δ 0 ) ) x ∈ Ω − , u ∗ ( x ) = ϑ ( c 2 K ( d ( x ) + δ 0 ) ) , x ∈ Ω + ,

where c 1 , c 2 are as in (5.8). Similarly, we can obtain that u ∗ is a viscosity supersolution of (1.10) in Ω − and u ∗ is a viscosity subsolution of (1.10) in Ω + .

Next, we want to establish the boundary asymptotic estimate of the blow-up solutions.

Let u be a positive viscosity solution of problem (1.10) in Ω . Note that the existence of u is guaranteed by Corollary 3.1 and Lemma 5.1. Now we will give the asymptotic estimate of u by the viscosity subsolution u ∗ and supersolution u ∗ based on the comparison principle. We note that

u ∗ ( x ) → ∞ , d ( x ) → δ 0 , and u > u ∗ on ∂ Ω .

Since ϑ and k are non-increasing (equality holds when k is non-decreasing), we can take a large constant M > 0 such that

u ≤ u ∗ + M , x ∈ { x ∈ Ω ∣ d ( x ) = δ }

and

u ∗ ≤ u + M , x ∈ { x ∈ Ω ∣ d ( x ) = δ − δ 0 } .

Therefore, u ≤ u ∗ + M on ∂ Ω − , and u ∗ ≤ u + M on ∂ Ω + . By the comparison principle, Theorem 2.1, we obtain

u ≤ u ∗ + M , in Ω − and u ∗ − M ≤ u , in Ω + .

Letting δ 0 → 0 , we obtain

u ∗ − M ≤ u ≤ u ∗ + M in Ω + ∩ Ω − .

That is,

ϑ ( c 2 K ( d ( x ) ) ) − M ≤ u ≤ ϑ ( c 1 K ( d ( x ) ) ) + M in Ω + ∩ Ω − .

Since ϑ ( K ( d ( x ) ) ) → ∞ as d ( x ) → 0 , by (5.5), we have

c 2 1 − α = lim d ( x ) → 0 ϑ ( c 2 K ( d ( x ) ) ) ϑ ( K ( d ( x ) ) ) ≤ lim inf d ( x ) → 0 u ( x ) ϑ ( K ( d ( x ) ) ) ≤ lim sup d ( x ) → 0 u ( x ) ϑ ( K ( d ( x ) ) ) ≤ lim d ( x ) → 0 ϑ ( c 1 ( K ( d ( x ) ) ) ) ϑ ( K ( d ( x ) ) ) = c 1 1 − α .

Passing to the limit as ε → 0 , we have

lim d ( x ) → 0 u ( x ) ϑ ( K ( d ( x ) ) ) = α + β − 1 α b 0 α − 1 h + 1 .

Now, we turn to prove the uniqueness of the viscosity solution when g ( t ) t h is non-decreasing on ( 0 , ∞ ) . We argue by contradiction. Suppose that u and v are two viscosity solutions of (1.10). Taking small δ > 0 , we can assume that u > 0 and v > 0 in Ω δ . And since ϑ ( d ( x ) ) → ∞ for d ( x ) → 0 , we can assume that u ∗ > 0 in Ω δ . Then we obtain

u ( x ) v ( x ) ≤ u ∗ ( x ) u ∗ ( x ) , x ∈ Ω δ ,

which implies that

lim sup x → ∂ Ω u ( x ) v ( x ) ≤ 1 .

By Theorem 2.2, we have u ≤ v in Ω . Similarly, by swapping u and v , we can obtain v ≤ u in Ω . The uniqueness result is proven.□

Remark 5.1

When α = 1 , g is rapidly varying at infinity and (1.11) reduces to

(5.14) lim d ( x ) → 0 u ( x ) ϑ ( K ( d ( x ) ) ) = 1 .

Therefore, relations (1.11) and (5.14) reveal the difference between the blow-up rate of the boundary blow-up solutions near the boundary of (1.10) when g is regularly varying at infinity ( α > 1 ) and rapidly varying at infinity ( α = 1 ). Furthermore, if g is rapidly varying at infinity, the blow-up rate does not depend on b 0 or β .

liufang78@njust.edu.cn

Acknowledgments

The authors would like to thank the anonymous referees for the careful reading of the manuscript and useful suggestions and comments.

Funding information: This work was supported by National Natural Science Foundation of China (No. 11501292).
Conflict of interest: The authors state no conflict of interest.

References

[1] G. Aronsson, Minimization problems for the functional supxF(x,f(x),f′(x)), Ark. Mat. 6 (1965), 33–53. 10.1007/BF02591326Search in Google Scholar

[2] G. Aronsson, Minimization problems for the functional supxF(x,f(x),f′(x)) II, Ark. Mat. 6 (1966), 409–431. 10.1007/BF02590964Search in Google Scholar

[3] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561. 10.1007/BF02591928Search in Google Scholar

[4] G. Aronsson, Minimization problems for the functional supxF(x,f(x),f′(x)) III, Ark. Mat. 7 (1969), 509–512. 10.1007/BF02590888Search in Google Scholar

[5] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), 51–74. 10.1007/BF00386368Search in Google Scholar

[6] S. Armstrong and C. Smart, An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differ. Equ. 37 (2010), 381–384. 10.1007/s00526-009-0267-9Search in Google Scholar

[7] M. G. Crandall, G. Gunnarsson, and P. Wang, Uniqueness of infty-harmonic functions and the eikonal equation, Comm. Partial Diff. Equ. 32 (2007), 1587–1615. 10.1080/03605300601088807Search in Google Scholar

[8] M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differ. Equ. 13 (2001), 123–139. 10.1007/s005260000065Search in Google Scholar

[9] G. Aronsson, M. G. Crandall, and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41 (2004), 439–505. 10.1090/S0273-0979-04-01035-3Search in Google Scholar

[10] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Adv. Math. 217 (2008), no. 4, 1838–1868. 10.1016/j.aim.2007.11.020Search in Google Scholar

[11] T. Bhattacharya and A. Mohammed, On solutions to Dirichlet problems involving the infinity-Laplacian, Adv. Calc. Var. 4 (2011), 445–487. 10.1515/acv.2010.019Search in Google Scholar

[12] T. Bhattacharya and A. Mohammed, Inhomogeneous Dirichlet problems involving the infinity-Laplacian, Adv. Differ. Equ. 17 (2012), no. 3–4, 225–266. 10.57262/ade/1355703086Search in Google Scholar

[13] E. N. Barron, R. R. Jensen, and C. Y. Wang, The Euler equation and absolute minimizers of L∞ functionals, Arch. Ration. Mech. Anal. 157 (2001), 255–283. 10.1007/PL00004239Search in Google Scholar

[14] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Diff. Equ. 26 (2001), 2323–2337. 10.1081/PDE-100107824Search in Google Scholar

[15] Y. Yu, Uniqueness of values of Aronsson operators and running costs in “tug-of-war” games, Annales de Linstitut Henri Poincare Non Linear Analysis 26 (2009), no. 4, 1299–1308. 10.1016/j.anihpc.2008.11.001Search in Google Scholar

[16] Y. Peres, O. Schramm, S. Sheffield, and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. 10.1007/978-1-4419-9675-6_18Search in Google Scholar

[17] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Part. Diff. Equ. 33 (2008), no. 10, 1788–1817. 10.1080/03605300802289253Search in Google Scholar

[18] G. Lu and P. Wang, Infinity Laplace equation with non-trivial right-hand side, Electron. J. Differ. Equ. 77 (2010), 517–532. Search in Google Scholar

[19] G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, Seminario Interdisciplinare di Matematica 7 (2008), 207–222. Search in Google Scholar

[20] E. N. Barron, L. C. Evans, and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. 360 (2008), 77–101. 10.1090/S0002-9947-07-04338-3Search in Google Scholar

[21] L. C. Evans, The 1-Laplacian, the infinity Laplacian and differential games, Contemp. Math. 446 (2007), 245–254. 10.1090/conm/446/08634Search in Google Scholar

[22] J. D. Rossi, Tug-of-war games and PDEs, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 2, 319–369. 10.1017/S0308210510000041Search in Google Scholar

[23] F. Liu, An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term, Commun Pure Appl. Anal. 17 (2018), no. 6, 2395–2421. 10.3934/cpaa.2018114Search in Google Scholar

[24] R. López-Soriano, J. C. Navarro-Climent, and J. D. Rossi, The infinity Laplacian with a transport term, J. Math. Anal. Appl. 398 (2013), 752–765. 10.1016/j.jmaa.2012.09.030Search in Google Scholar

[25] Y. Peres, G. Pete, and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. PDE. 38 (2010), no. 3–4, 541–564. 10.1007/s00526-009-0298-2Search in Google Scholar

[26] V. Caselles, J. M. Morel, and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process 7 (1998), no. 3, 376–386. 10.1109/ICIP.1997.632125Search in Google Scholar

[27] A. Elmoataz, M. Toutain, and D. Tenbrinck, On the p-Laplacian and infty-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci. 8 (2015), 2412–2451. 10.1137/15M1022793Search in Google Scholar

[28] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, 1–66. 10.1090/memo/0653Search in Google Scholar

[29] J. Garcia-Azorero, J. J. Manfredi, I. Peral, and J. D. Rossi, The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem, Nonlinear Analysis: Theory Meth. Appl. 66 (2007), 349–366. 10.1016/j.na.2005.11.030Search in Google Scholar

[30] P. Juutinen and J. D. Rossi, Large solutions for the infinity Laplacian, Adv. Calc. Var. 1 (2008), 271–289. 10.1515/ACV.2008.011Search in Google Scholar

[31] A. Mohammed and S. Mohammed, On boundary blow-up solutions to equations involving the ∞-Laplacian, Nonlinear Analysis Theory Meth. Appl. 74 (2011), 5238–5252. 10.1016/j.na.2011.04.022Search in Google Scholar

[32] A. Mohammed and S. Mohammed, Boundary blow-up solutions to degenerate elliptic equations with non-monotone inhomogeneous terms, Nonlinear Analysis Theory Meth. Appl. 75 (2012), 3249–3261. 10.1016/j.na.2011.12.026Search in Google Scholar

[33] W. Wang, H. Gong, and S. Zheng, Asymptotic estimates of boundary blow-up solutions to the infinity Laplace equations, J. Differ. Equ. 256 (2014), 3721–3742. 10.1016/j.jde.2014.02.018Search in Google Scholar

[34] Z. Zhang, Boundary behavior of large viscosity solutions to infinity Laplace equations, Z. Angew. Math. Phys. 66 (2015), 1453–1472. 10.1007/s00033-014-0470-1Search in Google Scholar

[35] Y. J. Chen, Boundary behavior for the large viscosity solutions to equations involving the infinity-Laplacian, Applicable Anal. 96 (2017), no. 12, 2065–2074. 10.1080/00036811.2016.1202405Search in Google Scholar

[36] L. Mi and C. Chen, The exact blow-up rate of large solutions to infinity-laplacian equation, J. Appl. Anal. Comput. 11 (2021), no. 2, 858–873. 10.11948/20200043Search in Google Scholar

[37] H. Wan, The exact asymptotic behavior of boundary blow-up solutions to infinity Laplacian equations, Zeitschrift für angewandte Mathematik und Physik 67 (2016), no. 4, 97. 10.1007/s00033-016-0694-3Search in Google Scholar

[38] H. Wan, The second order expansion of boundary blow-up solutions for infinity-Laplacian equations, J. Math. Anal. Appl. 436 (2016), 179–202. 10.1016/j.jmaa.2015.11.054Search in Google Scholar

[39] M. G. Crandall, H. Ishii, and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1–67. 10.1090/S0273-0979-1992-00266-5Search in Google Scholar

[40] F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased Tug-of-War, Adv. Nonlinear Stud. 19 (2019), no. 1, 89–112. 10.1515/ans-2018-2019Search in Google Scholar

[41] M. G. Crandall, A visit with the infty-Laplace equation, Calc. Var. Nonl. PDEs, Lecture Notes in Mathematics-Springer-verlag, 1927 (2008), 75–122. 10.1007/978-3-540-75914-0_3Search in Google Scholar

[42] C. C. Li, F. Liu, and P. B. Zhao, Boundary blow-up solutions to equation involved in infinity Laplacian, Under review. Search in Google Scholar

[43] F. Liu and X. P. Yang, Solutions to an inhomogeneous equation involving infinity-Laplacian, Nonlinear Anal Theory Meth Appl. 75 (2012), 5693–5701. 10.1016/j.na.2012.05.017Search in Google Scholar

[44] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. 10.1017/CBO9780511721434Search in Google Scholar

[45] E. Seneta, Regularly varying functions, Lecture Notes in Mathematics, vol. 508, Springer-Verlag, Berlin, 1976. 10.1007/BFb0079658Search in Google Scholar

[46] Y. J. Chen and M. X. Wang, Boundary blow-up solutions for p-Laplacian elliptic equations of logistic type, Proc. Royal Soc. Edingburg 142A (2012), 691–714. 10.1017/S0308210511000308Search in Google Scholar

Received: 2021-09-16

Revised: 2022-01-27

Accepted: 2022-01-28

Published Online: 2022-03-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

Large solutions of a class of degenerate equations associated with infinity Laplacian

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

Remark 1.1

Theorem 1.3

Theorem 1.4

2 Comparison principles

Definition 2.1

Definition 2.2

Lemma 2.1

Theorem 2.1

Proof

Remark 2.1

Theorem 2.2

Proof

Remark 2.2

3 Existence of boundary large solutions

Theorem 3.1

Proof

Remark 3.1

Remark 3.2

Lemma 3.1

Proof

Proof of Theorem 1.1

Corollary 3.1

Remark 3.3

4 Non-existence of boundary large solutions

Proof of Theorem 1.2

Proof of Theorem 1.3

5 Boundary asymptotic estimates

Definition 5.1

Definition 5.2

Proposition 5.1

Lemma 5.1

Proof of Theorem 1.4

Remark 5.1

Acknowledgments

References

Journal and Issue

Articles in the same Issue