Blow-up rates of large solutions for semilinearelliptic equations

In this paper we analyze the blow-up rates of large solutions to 
the semilinear elliptic problem $\Delta u =b(x)f(u), 
x\in \Omega, u|_{\partial \Omega} = +\infty,$ 
 where $\Omega$ is a bounded domain with smooth boundary 
in $R^N$, $f$ is rapidly varying or 
normalised regularly varying with index $p$ ($p>1$) at infinity, and $b \in 
C^\alpha (\bar{\Omega})$ which is non-negative in $\Omega$ and positive 
near the boundary and may be vanishing on the boundary.

1. Introduction and the main results. In this paper we analyze the blow-up rates of large solutions to the following semilinear elliptic problem u = b(x)f (u), x ∈ Ω, u| ∂Ω = +∞, (1.1) where the last condition means that u(x) → +∞ as d(x) = dist(x, ∂Ω) → 0, Ω is a bounded domain with smooth boundary in R N , b satisfies (b 1 ) b ∈ C α (Ω) for some α ∈ (0, 1), is non-negative in Ω; (b 2 ) there exists k ∈ Λ such that where Λ is the set of all positive non-decreasing functions in C 1 (0, δ 0 ) (δ 0 > 0) which satisfy lim t→0 + d dt We note that the set Λ was first introduced by Cîrstea and Rǎdulescu [6] for studying the boundary behaviour and uniqueness of solutions to problem (1.1) with the weight b satisfying (b 1 ) and the assumption that (b 3 ) there exist k ∈ Λ and b 0 > 0 such that The problem (1.1) arises from many branches of mathematics and has been discussed and extended by many authors in many contexts, for instance, the existence, boundary behavior and uniqueness of solutions, see, [1]- [4], [6]- [15], [17]- [26], [28], [29], [32]- [37] and the references therein.
In [37], the authors related the constants C f and C k and got the boundary behavior and uniqueness of solutions to problem (1.1) under (b 3 ).
Cîrstea and Rǎdulescu [6], Cîrstea and Du [7], Cîrstea [8] introduced the Karamata regular variation theory to study the boundary behavior and uniqueness of solutions to problem (1.1) and obtained a series of very rich significant information about the qualitative behavior of the boundary blow-up solutions in a general framework that removes previous restrictions in the literature.
Inspired by the above works, in this paper we obtained the boundary behavior and uniqueness of solutions to problem (1.1) under (b 2 ).
Our main results are summarized as the following. and (1.10) In particular, when C f = 1,  [20] [32] and [34]. Remark 1.3. By the following Lemmas 2.9 and 2.10, one can see that (1.7) is equivalent to the condition that (1.12) The outline of this paper is as follows. In Section 2, we need preliminary considerations. In Section 3 we prove Theorem 1.1 (I). The proof of Theorem 1.1 (II) is given in Section 4.

Preliminaries.
In this section, we present some bases of Karamata regular variation theory which come from Seneta [31], Preliminaries in Resnick [30], Introductions and the Appendix in Maric [27].
for some a > 0, is called regularly varying at infinity with index p, written as f ∈ RV p , if for each ξ > 0 and some p ∈ R, In particular, when p = 0, f is called slowly varying at infinity. (iii): e (ln s) q , 0 < q < 1.
We also say that a positive measurable function g defined on (0, a) for some a > 0, is regularly varying at zero with index p ( written as g ∈ RV Z p ) if t → g(1/t) belongs to RV −p . Similarly, g is called rapidly varying at zero if t → g(1/t) is rapidly varying at infinity. Proposition 2.3 (Uniform convergence theorem). If f ∈ RV p , then (2.1) holds uniformly for ξ ∈ [c 1 , c 2 ] with 0 < c 1 < c 2 . Moreover, if p < 0, then uniform convergence holds on intervals of the form (a 1 , ∞) with a 1 > 0; if p > 0, then uniform convergence holds on intervals (0, a 1 ] provided f is bounded on (0, a 1 ] for all a 1 > 0.

3)
for some a 1 ≥ a, where the functions ϕ and y are measurable and for s → ∞, We say thatL is normalised slowly varying at infinity and is normalised regularly varying at infinity with index p ( and written as f ∈ N RV p ). Similarly, g is said normalised regularly varying at zero with index p, written Proposition 2.5. If functions L, L 1 are slowly varying at infinity, then , are also slowly varying at infinity.
Proposition 2.7 (Asymptotic behaviour). If a function L is slowly varying at infinity, then for a ≥ 0 and t → ∞, Proposition 2.8 (Asymptotic behaviour). If a function H is slowly varying at zero, then for a > 0 and t → 0 + , Our results in this section are summarized as the following.

ZHIJUN ZHANG AND LING MI
Now let u be an arbitrary solution to problem (1.1). We assert that there exists a positive constant M such that where v 0 is the solution to problem (3.1).
In a similar way, we can show (3.7).
Hence, x ∈ D − ρ ∩ D + ρ , by letting ρ → 0, we have and . Consequently, and lim Thus by letting ε → 0, we have and lim By Lemma 2.11 (ii) and Proposition 2.3, we have In particular, when C f = 1, we have The proof is finished.
Finally, we prove the uniqueness of solutions to problem (1.1). Let u 0 be the minimal solution of problem (1.1), and let u be any other solution to problem (1.1). We prove u = u 0 in Ω. In fact, by the definition of the minimal solution, we have u 0 ≤ u in Ω. (4.16) Moreover, by the asymptotic behavior (1.11) and (4.1) we have For ε > 0 arbitrary, setting w : = (1 + ε)u 0 , we have We may assume that D ε is nonempty for ε small enough, for otherwise there is nothing to prove. Indeed, notice that D ε monotonically increases as ε ↓ 0. Moreover, we may also assume that D ε → Ω as ε → 0, for if there exists x 0 ∈ Ω and a sequence ε n → 0 such that x 0 ∈ D εn for all n, we have (1 + ε n )u 0 (x 0 ) ≥ u(x 0 ), and hence u 0 (x 0 ) = u(x 0 ). The strong maximum principle then yields u ≡ u 0 in Ω. Finally, we have D ε ⊂⊂ Ω by (4.18). Next we choose η > 0 so that u 0 ≥ S 0 in Ω η and define D ε,η = D ε ∩ D η . Notice that D ε,η is a non-empty open set for small ε. Moreover, we have by (4.3) that (4.20) It follows by (f 1 ) (or (f 01 )) that The maximum principle implies that u − u 0 ≤ θ in Ω η , and hence u − u 0 ≤ θ in the whole Ω. Then the strong maximum principle gives u − u 0 ≡ θ. We obtain that f (u) = f (u + θ) in Ω, which can only hold if θ = 0. Thus u ≡ u 0 , and this shows the uniqueness.