Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in Rn∖{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R^{n}\backslash\{0\}$\end{document}

We consider the following singular semilinear problem {Δu(x)+p(x)uγ=0,x∈D(in the distributional sense),u>0,in D,lim|x|→0|x|n−2u(x)=0,lim|x|→∞u(x)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$\end{document} where γ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma <1$\end{document}, D=Rn∖{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D=\mathbb{R}^{n}\backslash \{0\}$\end{document} (n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 3$\end{document}) and p is a positive continuous function in D, which may be singular at x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x=0$\end{document}. Under sufficient conditions for the weighted function p(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x)$\end{document}, we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.


Introduction and the main result
Semilinear elliptic partial differential equations of the type u(x) + p(x)u γ = 0 (1.1) will be considered in D = R n \{0} (n ≥ 3), where γ < 1 and p is a positive continuous function in D, which may be singular at x = 0. Our main goal is to establish sufficient conditions for the existence of a positive continuous solution u(x) of (1.1) with specified asymptotic behavior as |x| → 0 and as |x| → ∞. Global asymptotic behavior of this solution is also obtained. The importance of this type of equation in mathematics and applied mathematics has been widely recognized; see, for example, [11][12][13].
The above equation, subjected to homogeneous Dirichlet boundary conditions, has been intensively studied in the case where D = R n (n ≥ 3). In this sense, the existence of entire positive solutions for any γ < 0, that is the singular case, has been established by using the sub-supersolutions method in [26] or by other methods in [10]. These results have been extended to more general nonlinear terms, respectively, in [7,18,27], and [20].
In [4], the authors studied equation (1.1) on the whole space in the sublinear case. More precisely, they have proved the existence and uniqueness of the problem where 0 < γ < 1 and p is a nonnegative measurable function such that the function x → R n p(y) |x-y| n-2 dy belongs to L ∞ (R n ). In [5], by using Karamata regular variation theory and the sub-supersolutions method, the authors studied the asymptotic behavior as |x| → ∞ of the unique classical positive solution of problem (1.3) with γ < 1 and p(x) is a nonnegative function in C α loc (R n ), 0 < α < 1, such that there exists c > 0 satisfying where λ ≥ 2 and L belongs to the class of slowly varying functions at infinity (see Definition 1.1).
In [1], the authors considered equation (1.1) in a punctured bounded domain. Under some sufficient conditions on the function p(x), the existence of a positive continuous solution with a global behavior is obtained. Their approach is based on the Karamata regular variation theory and the Schauder fixed-point theorem.
The initial Karamata regular variation theory was developed by Karamata in [14]. In [8], the authors have shown that the class of Karamata regular variation functions is a well-suited framework for asymptotic analysis near the boundary for semilinear elliptic problems. For more works related to the Karamata regular variation theory, we refer the reader to [15-17, 19, 22, 24] and the reference therein.
Motivated by the approach used in [1] and [5], in this paper, we consider the existence and global asymptotic behavior of a positive continuous solution to the following problem where γ < 1, D = R n \{0} (n ≥ 3) and the potential function p(x) is required to satisfy some convenient comparable asymptotic rate related to the class of slowly varying functions defined as follows; see for example [2,14,21,25]: As examples, we quote: 1 3 cos(ln t) 1 3 }. The last example shows that the behavior at infinity for a slowly varying function cannot be predicted. Indeed, it exhibits "infinite oscillation" in the sense that On the other hand, the growth or decay of a slowly varying function as t → ∞ is limited in the sense that it satisfies for any ε > 0 Similarly, a class of normalized slowly varying (at zero) function is defined as follows: Definition 1.2 A positive continuously differentiable function L defined on (0, a], for some a > 0, is said to be normalized slowly varying (at zero) if, we write L ∈ N SV 0 . Throughout this paper, we make the following assumption: (H) p is a positive continuous function in D such that there exists c > 0 satisfying where P(x) := |x| -μ L 0 (min(|x|, 1))(|x| + 1) μ-λ L ∞ (max(|x|, 1)), with γ < 1, μ ≤ n + (2n)γ and λ ≥ 2.
Here, L 0 ∈ N SV 0 , defined on (0, a], for some a > 1 and (1.8) Note that the comparable asymptotic rate of p(x) in (1.7) determines the asymptotic behavior of the solution.
Our main result is summarized in the following theorem.
where c is a positive constant and for x ∈ D, and (1.12) Remark 1.5 From (1.9) and (1.6), we obtain That is, the solution blows-up at the origin.
The outline of this article is as follows. In Sect. 2, we prove some pertinent properties related to the Kato class and also to the Karamata regular variation theory. In Sect. 3, we show the existence of a solution to problem (1.5) with the required asymptotic behavior (1.9).
In this paper, we use the following notations: Note that 0 ∈ S + (R n ) and harmonic on D, see, for example, [3]. (viii) For x, y ∈ R n , we denote the normalized fundamental solution of Laplace's equation by: (1.14) From [6, Proposition 2.10], we learned that if f ∈ B + (D) such that f ∈ L 1 loc (D) and N f ∈ L 1 loc (D), then Throughout this paper, the letter c will denote a generic positive constant that may vary from line to line.

Kato class K
where (x, y) is given by (1.13).
The next Lemma is due to Mâagli and Zribi, see [20, Remark 2 and Proposition 1].
Therefore, we need to prove (2.4) and (2.5) only for h(y) = (y, z) uniformly in z ∈ D.
Let r > 0. By using Remark 2.4, there exists a constant c > 0, such that for all x, y, z ∈ D, For ε > 0, by Definition 2.1, there exists s > 0 and M > 0 such that Using this fact, (2.6) and Lemma 2.3(ii), we obtain (2.4) by letting r → 0. Finally, note that assertion (2.5) follows by using similar arguments as above.
Proof Let ψ ∈ K ∞ n (D) and x 0 ∈ R n . Since 0 ∈ S + (D), then for ε > 0, by Proposition 2.5, there exists M > r > 0, such that the following holds: , we obtain by Lemma 2.3 (ii) and Lebesgue's dominated convergence theorem, Hence, there exists δ > 0 with δ < r 2 such that if x ∈ B(x 0 , δ) ∩ D, That is, where c is some positive constant.
The following result concerns operations that preserve slow variation.
The following is an analog of Proposition 2.9 for L defined at zero instead of ∞. The following result, will play a central role in establishing our main result in Sect. 3.

Proof of Theorem 1.4
In order to prove Theorem 1.4, we need to establish some preliminary results. Our approach is inspired from methods developed in [20] with necessary modifications. For ν > 0, we denote by (P ν ) the following problem lim |x|→0 |x| n-2 u(x) = ν, We recall that for x ∈ D, 0 (x) = 1+|x| n-2 |x| n-2 .
Note that ν 0 (x) is a solution of the following homogeneous problem Problem (P ν ) can be seen as a perturbation of problem (H ν ). In particular, Proof Let γ < 0 and ν > 0. Due to Lemma 2.3 (i) and hypothesis (H), the function ψ(y) := ( 0 (y)) (γ -1) p(y) becomes in K ∞ n (D). Therefore, by Proposition 2.6, we have Let β 0 := ν + ν γ h ∞ and consider the convex set given by Define the operator T on by Since for all ϑ ∈ , ϑ γ ≤ ν γ , then as in the proof of Proposition 2.6 we show that the family T is equicontinuous in R n ∪ {∞}. In particular, for all ϑ ∈ , Tϑ ∈ C(R n ∪ {∞}) and so T ⊂ .
To prove the continuity of T in , we consider a sequence (ϑ k ) k ⊂ and ϑ ∈ such that ϑ kϑ ∞ → 0 as k → ∞. Then, we have we deduce by the dominated convergence theorem and Proposition 2.6 that ∀x ∈ R n ∪ {∞}, Tϑ k (x) → Tϑ(x) as k → ∞.
By using (3.15), we obtain for all v ∈ E, For all v ∈ E, we have v γ (y) ≤ c γ ω γ ∞ , for all y ∈ D.
Therefore, as in the proof of Proposition 2.6, we deduce that Tv ∈ C 0 R n , for all v ∈ E.
So, by the convergence monotone theorem, the sequence (ω k ) k converges to a function v satisfying for each x ∈ D, and v(x) = 1 0 (x) D (x, y)p(y) 0 (y) γ v γ (y) dy.
Since v is bounded, we prove by similar arguments as in the proof of Proposition 2.6 that v ∈ C 0 (R n ).