Singular solutions of a nonlinear elliptic equation in a punctured domain

We consider the following semilinear problem  −∆u(x) = a(x)uσ(x), x ∈ Ω\{0} (in the distributional sense), u > 0, in Ω\{0}, lim |x|→0 |x|n−2 u(x) = 0, u(x) = 0, x ∈ ∂Ω, where σ < 1, Ω is a bounded regular domain in Rn (n ≥ 3) containing 0 and a is a positive continuous function in Ω\{0}, which may be singular at x = 0 and/or at the boundary ∂Ω. When the weight function a(x) satisfies suitable assumption related to Karamata class, we prove the existence of a positive continuous solution on Ω\{0}, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.


Introduction
Let Ω be a bounded C 1,1 -domain in R n (n ≥ 3) containing 0. In [33], Zhang and Zhao proved the existence of infinitely many positive solutions for the following superlinear elliptic where B(Ω) be the set of Borel measurable functions in Ω.
In the present paper, we are interested in the singular and sublinear case.More precisely, we are concerned with the existence and global behavior of positive continuous solutions to the following nonlinear problem: where σ < 1 and a is a positive continuous function in Ω\{0} which may be singular at x = 0 and/or at the boundary ∂Ω.The weight function a(x) is required to satisfy suitable assumptions related to the following Karamata class K. Definition 1.1.Let η > 0 and L be a function defined on (0, η).Then L belongs to the class K if where c > 0 and v ∈ C([0, η]) with v(0) = 0.
Remark 1.2.This definition implies that the class K is given by We refer to [2,25,29] for examples of functions belonging to the class K.A class of functions in this class is defined by and ω is a sufficiently large positive real number such that L is defined and positive on (0, η), and are frequently used as some weight functions (see, for example, [17] and [19]).
Observe that functions belonging to the class K are in particular slowly varying functions.The initial theory of such functions was developed by Karamata in [16].
In [7], Cîrstea and Rȃdulescu have proved that the Karamata theory is very useful to study the asymptotic analysis of solutions near the boundary for large classes of nonlinear elliptic problems.
Throughout this paper, we assume that (H) a is a positive continuous function in Ω\{0} satisfying where We introduce the function θ defined in Ω\{0} by where L 1 and L 2 are defined on (0, η) by and s ds, if λ = 2. Using Karamata's theory and the Schauder fixed point theorem, we prove the following qualitative property.Theorem 1.3.Let σ < 1 and assume that the function a satisfies (H).Then problem (1.2) has at least one positive continuous solution u on Ω\{0} such that for x ∈ Ω\{0}, where c is a positive constant.
From now on, we denote by B + (Ω) the collection of all nonnegative Borel measurable functions in Ω.We refer to the set C(Ω) of all continuous functions in Ω and let C 0 (Ω) be the subclass of C(Ω) consisting of functions which vanish continuously on ∂Ω.For f , g ∈ B + (Ω) , we say that f ≈ g in Ω, if there exists c > 0 such that 1 c f (x) ≤ g(x) ≤ c f (x), for all x ∈ Ω.The letter c will denote a generic positive constant which may vary from line to line.
We define the potential kernel V on B + (Ω) by We recall that for any function Note that for any function f ∈ B + (Ω) such that V f (x 0 ) < ∞ for some x 0 ∈ Ω, we have [6,Lemma 2.9]).

Preliminaries and key tools 2.1 Green's function
In this section, we recall some basic properties on G(x, y), the Green's function of the Laplace operator in Ω.By [32], we have where σ is the normalized measure on the unit sphere S n−1 of R n .
The next result is due to Mâagli and Zribi, see [22,Lemma 1].

Kato class K(Ω)
In this subsection, we recall and prove some properties concerning the class K(Ω).

Karamata class
In this section, we collect some properties of the Karamata functions, which will be used later.
(i) For L ∈ K and ε > 0, we have (ii) Let L 1 , L 2 ∈ K and p ∈ R. Then we have (iii) For L ∈ K, we have In particular, we have s ds < ∞, then we have lim In particular The following properties are equivalent.
Proof.The proof follows by similar arguments as in [24, Proposition 7].
Next, we recall the following lemma due to Lazer and McKenna [18,p. 726].
Following the proof of the previous lemma, we deduce the following property.
(2.5) Now, it is clear that the function is positive and continuous on Ω\{0}.
On the other hand, by using (2.3) and Proposition 2.8, the function belongs to the class K(Ω).So, observing that b(x) = |x| 2−n q(x), we deduce by Proposition 2.5 that the function Vb is positive and continuous on Ω\{0}.
Proof.Let a be a function satisfying (H).Using (1.3) and (1.5), we obtain and λ ≤ 2, then one can easy check that γ ≤ n and ν ≤ 2. Now using Lemmas 2.6 and 2.7 and Proposition 2.12 with This completes the proof.

Proof of Theorem 1.3
This section is devoted to the proof of Theorem 1.3.So, we need to establish some preliminary results.Our approach is inspired from methods developed in [22,24] with necessary modifications.
By (3.4), we have lim This completes the proof.

Since for each
then u is upper and lower semi-continuous function on Ω\{0} and so u ∈ C(Ω\{0}).
Finally, using the fact that for all x ∈ Ω\{0}, 0 < u(x) ≤ u k (x) and that u k is a solution of problem (P α k ) , we deduce that lim |x|→0 |x| n−2 u(x) = 0 and lim Hence u is a solution of problem (1.2).

Proof of Theorem 1.3
Assume that the function a satisfies hypothesis (H).By Proposition 2.13, there exists M ≥ 1 such that for each Ω\{0}, where θ is the function defined in (1.5) and p(y) := a(y)θ σ (y).
To prove Theorem 1.3, we discuss the following two cases.
Similarly, we prove that 1 c V p ≤ u.
Thus, by (3.12) u satisfies (1.6).By using (3.17), we obtain for all v ∈ A, Since for all v ∈ A, we have Since the operator T is nondecreasing and T (A) ⊂ A, we deduce that Therefore, the sequence (v k ) k converges by the convergence monotone theorem to a func- tion v satisfying for each x ∈ Ω 1 c ϕ(x) ≤ v(x) ≤ cϕ(x) and v(x) = |x| n−2 Ω G(x, y)a(y) |y| (2−n)σ v σ (y)dy.
Since v is bounded, we prove by similar arguments as in the proof of Proposition 3.1 that v ∈ C 0 (Ω).

and L p 1
are in K.