Exact Boundary Behavior for the Solutions to a Class of Infinity Laplace Equations

In this paper, by Karamata regular variation theory and the method of lower and upper solutions, we give an exact boundary behavior for the unique solution near the boundary to the singular Dirichlet problem −∆ ∞ u = b(x)g(u), u > 0, x ∈ Ω, u| ∂Ω = 0, where Ω is a bounded domain with smooth boundary in R N , g ∈ C 1 ((0, ∞), (0, ∞)), g is decreasing on (0, ∞) and the function b ∈ C(¯ Ω) which is positive in Ω. We find a new structure condition on g which plays a crucial role in the boundary behavior of the solutions.


Introduction and the main results
Let Ω be a bounded domain with smooth boundary in R N (N ≥ 2).In this paper, we consider the exact asymptotic behavior near the boundary to the following singular Dirichlet problem where the operator ∆ ∞ is the ∞-Laplacian, and it is defined as b satisfies and g satisfies (g 1 ) g ∈ C 1 ((0, ∞), (0, ∞)), lim s→0 + g(s) = ∞ and g is decreasing on (0, ∞).
B Corresponding author.Email: mi-ling@163.comThis operator (1.2) is called the infinity Laplacian, which was first introduced in the work of Aronsson [2] in connection with the geometric problem of finding the so-called absolutely minimizing functions in Ω.As a result of the high degeneracy of the ∞-Laplacian, the associated Dirichlet problems may not have classical solutions.Therefore solutions are understood in the viscosity sense, a concept introduced by Crandall, Lions [13] and Crandall, Evans, Lions [12], and to be defined in Section 2. By using the viscosity solutions, Jensen [21] proved the existence and uniqueness of the viscosity solutions to the Dirichlet problem to the infinity harmonic equation.Later, Lu and Wang [23] obtained a uniqueness theorem for the Dirichlet problem to the infinity harmonic equation in the perspective of PDE.The infinity Laplace equation in turn is a very topical differential operator that appears in many contexts and has been extensively studied, see, for instance, [3, 5, 6, 11, 24-26, 29, 32, 33, 40] and the references therein.
Next, let us review the following singular elliptic boundary value problem involving the classical Laplace operator ∆, i.e.
Problem (1.3) arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrical materials and has been discussed and extended by many authors in many contexts, for instance, the existence, uniqueness, regularity and boundary behavior of solutions, see, [1, 4, 14-20, 22, 28, 30, 34, 35, 41-43, 45, 46] and the references therein.
It is very worthwhile to point out that Cîrstea and Rǎdulescu [8][9][10] introduced the Karamata regular variation theory which is a basic tool in stochastic process to study the boundary behavior and uniqueness of solutions to boundary blow-up elliptic problems and obtained a series of rich and significant information about the boundary behavior of solutions.For further insight on the boundary blow-up elliptic problems, please refer to [36,37,44] and the references therein.
Later, by means of Karamata regular variation theory, Zhang et al. [42,43,45,46] proved the first or second boundary expansion of solutions to problem (1.3).The author et al. [28,30] further proved the second boundary expansion of solutions to problem (1.3).
Ben Othman, Maagli, Masmoudi, Zribi [4] and Gontara, Maagli, Masmoudi, Turki [19] introduced a large class of functions b(x) which belong to the Kato class K(Ω) and proved the boundary behavior of solutions for problem (1.1) when g is normalized regularly varying at zero with index −γ (γ > 0).Later, Zhang et al. [45] extend the previous results on the boundary behavior of the solution u of problem (1.1) to the case where the weight functions b(x) belong to the Kato class K(Ω) or b(x) lie into a class of functions Λ that was introduced by Cîrstea and Rǎdulescu in [8][9][10] for non-decreasing functions and by Mohammed in [31] for nonincreasing functions as the set of positive monotonic functions Recently, N. Zeddini et al. [41] gave a common proof for theorems in [45] and extended these results.Now let us return to problem (1.1).
When Ω is a bounded domain that satisfies both the uniform interior and uniform exterior sphere conditions and b ≡ 1 in Ω, Bhattacharya and Mohammed [5] established that: let g satisfy (g 1 ) and u be a solution of (1.1), then there are positive constants a and c, with 0 < a < c such that where d(x) is the distance of x from ∂Ω, and Recently, the author [29] extended the result in [5] to the weight function b which belong to the set Λ. Theorem 1.1 in [29] established the following result: let g satisfy (g 1 ) and (g 2 ) there exists γ > 1 such that lim If C k (γ + 3) > 4, then for the unique solution u of problem (1.1), it holds that lim where φ is uniquely determined by and . (1.9)

L. Mi
For convenience, we introduce the following class of functions.
Let Λ 1 denote the set of all Karamata functions L, which are normalized slowly varying at zero defined on (0, a) for some a > 0 by for some a 1 ∈ (0, a), where c 0 > 0 and the function y ∈ C((0, a 1 ]) with lim s→0 + y(s) = 0. Inspired by the above works, in this paper, by Karamata regular variation theory and the method of lower and upper solutions, we investigate the new boundary asymptotic behavior of solutions to problem (1.1) when the weight function b lies into Λ 1 and the nonlinear term g satisfies the following structure condition Note that in this paper we extend the previous results in all two directions.We extend g(u) to a more general class of functions which include the condition (g 2 ) and b(x) belongs to another class of functions Λ 1 .
Our main results are summarized as follows.
where φ is uniquely determined by (1.8), and The outline of this paper is as follows.In Sections 2-3, we give some preparation that will be used in the next section.The proof of Theorem 1.1 will be given in Section 4.

Preparation
Our approach relies on Karamata regular variation theory established by Karamata in 1930 which is a basic tool in the theory of stochastic process (see [7,27,39] and the references therein).In this section, we first give a brief account of the definition and properties of regularly varying functions involved in our paper (see [7,27,39]).Definition 2.1.A positive measurable function f defined on [a, ∞), for some a > 0, is called regularly varying at infinity with index ρ, written as f ∈ RV ρ , if for each ξ > 0 and some ρ ∈ R, In particular, when ρ = 0, f is called slowly varying at infinity.
We also see that a positive measurable function g defined on (0, a) for some a > 0, is regularly varying at zero with index σ (written as g ∈ RVZ σ ) if t → g(1/t) belongs to RV −σ .Similarly, g is called rapidly varying at zero if t → g(1/t) is rapidly varying at infinity.

L. Mi
Proposition 2.3 (Uniform convergence theorem).If f ∈ RV ρ , then (2.1) holds uniformly for ξ ∈ [c 1 , c 2 ] with 0 < c 1 < c 2 .Moreover, if ρ < 0, then uniform convergence holds on intervals of the form (a 1 , ∞) with a 1 > 0; if ρ > 0, then uniform convergence holds on intervals (0, a 1 ] provided f is bounded on (0, a 1 ] for all a 1 > 0. Proposition 2.4 (Representation theorem).A function L is slowly varying at infinity if and only if it may be written in the form for some a 1 ≥ a, where the functions ϕ and y are measurable and for s → ∞, y(s) → 0 and ϕ(s) → c 0 , with c 0 > 0.
We say that is normalized slowly varying at infinity and is normalized regularly varying at infinity with index ρ (and written as f ∈ NRV ρ ).Similarly, g is called normalized regularly varying at zero with index ρ, written as g ∈ NRVZ ρ if t → g(1/t) belongs to NRV −ρ .
in Ω if for every ϕ ∈ C 2 (Ω), with the property that u − ϕ has a local maximum at some in Ω if for every ϕ ∈ C 2 (Ω), with the property that ū − ϕ has a local minimum at some in Ω if it is both a subsolution and a supersolution.

Some auxiliary results
In this section, we collect some useful results that will be used in the proof of the theorem.
(i) If g satisfies (g 2 ), then C g ≤ 1; (ii) (g 2 ) holds for C g ∈ (0, 1) if and only if g ∈ NRV −γ ; with γ > 0. In this case γ = 3C g /(1 − C g ); (iii) (g 2 ) holds for C g = 0 if and only if g is normalized slowly varying at zero; (iv) if (g 2 ) holds with C g = 1, then g is rapidly varying to infinity at zero; then g satisfies (g 2 ) with C g = 1.
In a similar way, we can show that is a subsolution of equation (1.1) in Ω δ 1ε .Let u ∈ C(Ω) be the unique solution to problem (1.1).We assert that there exists M large enough such that u(x) ≤ Mv(x) + ūε (x), u ε (x) ≤ u(x) + Mv(x), x ∈ Ω δ 1ε , ( where v is the solution of problem (4.1).

(b 2 )
there exist some k ∈ Λ and a positive constant b 0 ∈ R such that lim
and consequently ∆ ∞ d = 0 in Ω δ 1 in the viscosity sense.Proof of Theorem 1.1.Let v ∈ C( Ω) be the unique solution of the problem