Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian

<jats:p>The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mo stretchy="false">{</mml:mo><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi mathvariant="normal">′</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">lim</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>→</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">′</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">3</mml:mn><mml:mo>,</mml:mo></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>σ</mml:mi><mml:mo><</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>,</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:math> denotes a nonnegative continuous function that might have the property of being singular at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math> and /or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math> and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math>, while the solution could blow-up at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:math>. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.</jats:p>

Let us first introducing the following functional class K called Karamata class.
Here, it is pertinent to note that the functions in the class K are slowly varying, and Karamata developed in [31] the initial theory in this field.
Cirstea and Rȃdulescu have exploited in [32] the Karamata theory to study the asymptotic and qualitative behavior near the boundary of solutions of nonlinear elliptic problems.
The aim of this paper is to address the existence, uniqueness and qualitative behavior of positive continuous solution to the following singular nonlinear problem.
where ≥ 3, < 1 and denotes a nonnegative continuous function on (0, 1) that might have the property of being singular at = 0 and /or = 1 and which might satisfies 2 Journal of Function Spaces certain condition associated with the Karamata class K. In this situation, the nonlinearity might also have the property of being singular at = 0. Here we emphasizes that the obtained solution may also blow-up at 0, which is not given in the previous works. Our approach relies on Karamata theory and the Schauder fixed point theorem.
, is the set of continuous (resp. nonnegative continuous) functions in a metric space .
is given by Clearly the solution blow up at 0 for ≥ 2 and also we have This implies that the global estimates obtained in Theorem 2 are optimal.

Karamata Class.
It is clear to see that for some > 1, the class K is given by Standard examples of functions which are elements of the class K are presented below (see [33][34][35]) (log ( )) , where log = log ∘ log ∘ ⋅ ⋅ ⋅ log ( times), ∈ R, ] ∈ (0, 1) and is a sufficiently large positive real number such that is defined and positive on (0, ], for some > 1.
Next, we collect several properties of the Karamata functions, which will be useful in the proof of our main result.

Global Estimates.
Let ≥ 3, then is the Green's function of the operator ( ) There exists a constant > 0 such that for all , ∈ [0, 1], Proof. (i) The property follows from (21) and the fact that (ii) The inequalities follow from (22) and the fact that For ∈ B + ((0, 1)), we define the otential of by Using Lemma 7, we deduce the following.
By simple calculation, we obtain That is By differentiating (31), we obtain for ∈ (0, 1), Using (31) and the fact that 4

Proof of Theorem 2.
Assume that hypothesis ( ) is fulfilled. Let be the function defined in (7). By Proposition 11, there exists ≥ 1 such that for each ∈ (0, 1), where ( ) fl ( ) ( ). We will break up the proof in two cases.

Journal of Function Spaces
Since the operator F is nondecreasing and F( ) ⊂ , we deduce that Therefore, the sequence (V ) converges by the convergence monotone theorem to a function V satisfying for each ∈ Which implies by Proposition 9 that 0 V − = ( ( 0 V − )) ≥ 0. By symmetry, we obtain that 0 ≥ V. Hence, 0 ∈ . Using the fact that < 1, we get 0 = 1. Then, we conclude that = V.
This completes the proof of Theorem 2.