Asymptotic behavior and uniqueness of entire large solutions to a quasilinear elliptic equation

In this paper, combining the upper and lower solution method with perturbation theory, we study the asymptotic behavior of entire large solutions to Eq. ∆pu = b(x) f (u), u(x) > 0, x ∈ R, where b ∈ Cα loc(R N) (α ∈ (0, 1)) is positive in RN (N ≥ 3), f ∈ C1[0, ∞) is positive on (0, ∞) which satisfies a generalized Keller– Osserman condition and is rapidly varying or regularly varying with index μ ≥ p− 1. We then discuss the uniqueness of solutions by the asymptotic behavior of entire large solutions at infinity.


Introduction
In this article, we study the exact asymptotic behavior of entire large solutions u ∈ W 1,p loc (R N ) ∩ C 1,α  loc (R N ) (α ∈ (0, 1)) to the following quasilinear elliptic equation where ∆ p u := div(|∇u| p−2 ∇u) stands for p -Laplacian operator with 1 < p < N (N ≥ 3).The entire large solution means that u solve Eq. (1.1) in R N and u(x) → ∞ as |x| → ∞, which is also called "entire blow-up solution" or "entire explosive solution" in many different contexts.
The nonlinearity f satisfies the following hypotheses: 0) = 0, f (t) ≥ 0 and f (t) > 0 for t > 0; (f 2 ) the following generalized Keller-Osserman condition holds,  For p = 2, Eq.(1.1) has been extensively investigated by many authors and the link between Eq. (1.1) and geometric problem has been known for a long time, for instance, when b ≡ 1 in Ω, f (u) = e u and N = 2, Bieberbach [7] first analyzed the existence, uniqueness and asymptotic behavior of boundary blow-up solutions to Eq. (1.1) with p = 2 in a bounded domain Ω ⊆ R N with C 2 -boundary.In the case, Eq. (1.1) plays an important role in the theory of Riemannian surfaces of constant negative curvatures and in the theory of automorphic functions.Later, Rademacher [40], using the ideas of Bieberbach, extended the results to a bounded domain in R 3 .On the other hand, when f (u) = u γ , γ = (N + 2)/(N − 2), Yamabe [45] showed the relationship between solvability of Eq. (1.1) with p = 2 and the existence of a conformal metric on the Euclidean space R N , with a prescribed scalar curvature.It is worth while to point out that, Keller [25] and Osserman [39] carried out a systematic research on Eq. (1.1) with p = 2 and gave, respectively, the necessary and sufficient condition for the existence of large solutions when b ≡ 1 in bounded domain Ω and b ≡ 1 in R N .Then Lazer and McKenna [29], Lair [26][27][28], Cîrstea and Rȃdulescu [10], further investigate the existence of large solutions to Eq. (1.1) in bounded and unbounded domains.
When b ≡ 1 in a bounded domain Ω, the existence of large solutions to Eq. (1.1) was first studied by Diaz and Letelier [16] for f (u) = u γ (γ > p − 1).Then, Matero [33] studied the existence and asymptotic behavior of large solutions to Eq. (1.1) in a bounded smooth domain with a C 2 -boundary.If b ≡ 1 in bounded domain Ω ⊆ R N and f is a smooth, positive, and increasing function which satisfies (f 2 ), Gladiali and Porru [21] showed that if F(t)t −p is increasing for large t, then any weak solution u to problem (1.1) satisfies Furthermore, they showed that, under the additional assumption If b is non-negative and continuous on a bounded domain Ω ⊆ R N and satisfies some appropriate additional condition, Mohammed [36] established the existence and asymptotic behavior of large solutions to Eq. (1.1).Then, when b satisfies some suitable integral condition and p ∈ (1, N) (N ≥ 2), Covei [15] studied the existence of entire large solutions to Eq. (1.1) in R N .On the other hand, for the cases of f (u) = u γ with γ > p − 1 and f (u) = e u , García-Melián [19,20] investigated, respectively, the existence, uniqueness and asymptotic behavior of boundary blow-up solutions to Eq. (1.1) in a smooth bounded domain.
In different direction, by applying Karamata regular variation theory Cîrstea and Rȃdulescu [11][12][13][14] opened up a unified new approach to studied the boundary behavior and uniqueness of large solutions to Eq. (1.1) with p = 2 in a bounded domain, which enables us to obtain some significant information about the qualitative behavior of large solutions in a general framework.Later, Mohammed [37], Zhang et al. [46], Zhang [47][48][49], Huang et al. [23], Huang [24], Mi et al. [34], Mi and Liu [35] apply similar techniques to further study asymptotic behavior and uniqueness of boundary blow-up solutions to (1.1) in a bounded domain Ω ⊆ R N .Most recently, inspired by the above works, we [44] investigated the asymptotic behavior of entire large solutions to Eq. (1.1) with p = 2 in R N by using Karamata regular variation theory.
In this paper, we investigate the exact asymptotic behavior and uniqueness of entire large solutions to (1.1) in R N .Let f satisfy (f 1 )-(f 2 ), ψ be the solution of (1.3), we conclude by Lemmas 3.1 and 3.2 (v) that (iii ) holds with C f = 0 and in the case, ψ is rapidly varying to infinity at zero (please refer to Definition 2.3).
Our results are summarized as follows.
where ψ is uniquely determined by (1.3) and where and further satisfy the condition that where β > 1.Then a direct calculation shows that C f = 0 and (2) Let on (e, ∞), which satisfies F and F are increasing on [0, e] and In the case, a simple calculation shows that we arrive at where β ≥ 2. In the case, a straightforward calculation shows that C f = 0 and The paper is organized as follows.In Section 2, we give some bases of Karamata regular variation theory.In Section 3, we collect some preliminary considerations.The proof of Theorem 1.1 is given in Section 4. Finally, Section 5 is devoted to prove the uniqueness of entire large solutions.

Some basic facts from Karamata regular variation theory
In this section, we introduce some preliminaries of Karamata regular variation theory which come from [32,41,42].Definition 2.1.A positive continuous function f defined on [a, ∞), for some a > 0, is called regularly varying at infinity with index µ, denoted by f ∈ RV µ , if for each ξ > 0 and some µ ∈ R, In particular, when µ = 0, f is called slowly varying at infinity.
Clearly, if f ∈ RV µ , then L(t) := f (t)/t µ is slowly varying at infinity.We also see that a positive continuous function h defined on (0, a) for some a > 0, is regularly varying at zero with index µ (written as

Auxiliary results
In this section, we collect some useful results.
(iv) If (f 3 ) holds with C f = 1/p, then f is rapidly varying to infinity at infinity.
Proof.(i) Let Integrate J from a > 0 to t > a and integrate by parts, we obtain that It follows by L'Hospital's rule that i.e., lim (ii) (Necessity.)By (3.1) and L'Hospital's rule, we have

H. Wan
So, F ∈ NRV p/(1−pC f ) , i.e., there exist a large constant t 0 > 0 and a slowly varying function at infinity L ∈ C 2 [t 0 , ∞) such that where Furthermore, we have (Sufficiency).Let By Lebesgue's dominated convergence theorem, we have This implies that F ∈ p/(1 − pC f ).On the other hand, by using reduction to absurdity we can see that lim t→∞ (F(t)) On the other hand, by using L'Hospital's rule, we have We conclude by (3.6)-(3.7)that (f 3 ) holds with C f = 0.
(iv) When C f = 1/p, from the similar calculation as (3.3), we can see that So, for an arbitrary γ > 1, there exists t 0 > 0 such that Integrating the above inequality from t 0 to t, we obtain ln Letting t → ∞, the Definition 2.2 shows that F is rapidly varying at infinity.This combined with (f 1 ) shows that f is also rapidly varying at infinity.
Proof.(i) By the definition of ψ and a straightforward calculation, we can show that (i)-(ii) holds. (iii) (iv) By using (3.1) and L'Hospital's rule, we obtain lim (v) It follows by the similar calculation as (3.9) that lim Hence, for an arbitrary γ > 0, there exists a small enough t 0 > 0 such that Integrate it from t to t 0 , we obtain that ln ψ(t) − ln ψ(t 0 ) > (1 + γ)(ln t 0 − ln t), t ∈ (0, t 0 ], i.e., Letting t → 0 + , we see by Definition 2.3 that ψ is rapidly varying to infinity at zero.

Lemma 3.4 (Weak comparison principle).
Let Ω be a bounded domain and G : Ω × R → R be non-increasing in the second variable and continuous.Let u, w ∈ W 1, p (Ω) satisfy the respective inequalities for all non-negative ϕ ∈ W 1, p 0 (Ω).Then the inequality u ≤ w on ∂Ω implies u ≤ w in Ω.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1.
From (b 1 )-(b 2 ), Proposition 2.6 and Lemma 3.2 (iii), we see that corresponding to ε, there exist sufficiently small δ ε > 0 and large enough R ε > 0 such that for any (x, r) ∈ Ω R ε × (0, 2δ ε ), and and let u be an arbitrary entire large solution of Eq. (1.1).Define , where and We may as well assume that and set A straightforward calculation combined with (4.1) and (4.2) shows that for any This implies that u ε is a supersolution of Eq. (1.1) in D σ − .In a similar way, we can show that u ε is a subsolution of Eq. (1.1) in D + σ .We assert that there exists a large constant M > 0 independent of σ such that In fact, we can choose a positive constant M independent of σ such that when Moreover, we also see This implies that, we can take a sufficiently small ρ > 0 such that sup where Combining (4.6) with (4.8), we have On the other hand, we conclude by (4.7) and the definition of Ω + σ (please refer to (4.3)) that We note that u and u ε both satisfy (3.10)    .

Proof of Theorem 1.2
Proof.The existence of entire large solutions follows from Theorem 1.3 of [15].Inspired by the ideas of Mohammed in [37], we prove the uniqueness.Suppose So, for fixed ε > 0, there exists a large constant R ε such that By using (f 4 ), we obtain Let u 0 is the unique solution of where Ω 0 = R N \ Ω R ε .We conclude by Lemma (3.4) that u − (x) ≤ u 0 (x) ≤ u + (x), x ∈ Ω 0 . (5.2) Noting u 0 = u 1 on Ω 0 , so it follows by combining (5.1) with (5.2) that Letting ε → 0, we obtain u 1 = u 2 in R N .

u 1 and u 2
are entire large solutions of problem (1.1).It follows by Theorems 1 then f is rapidly varying to infinity at infinity (please refer to Definition 2.2);

Definition 2.2. A
positive continuous function f defined on [a, ∞), for some a > 0, is called rapidly varying to infinity at infinity ifIf f ∈ RV µ , then (2.1) holds uniformly for ξ ∈ [c 1 , c 2 ] with 0 < c 1 < c 2 .forsomea 1 ≥ a, where the functions ϕ and y are continuous and for t → ∞, y(t) → 0 and ϕ(t) → c 0 , with c 0 > 0. If ϕ ≡ c 0 , then L is called normalized slowly varying at infinity andf (t) = t µ L(t), t ≥ a 1 ,is called normalized regularly varying at infinity with index µ (written as f ∈ NRV µ ).
in Dσ − and D σ + , respectively.Moreover, by (f 1 ) we obtain that u ε + M and u + M are both supersolutions in Dσ