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Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in \(R^{n}\backslash\{0\}\)
Boundary Value Problems volume 2022, Article number: 4 (2022)
Abstract
We consider the following singular semilinear problem
where \(\gamma <1\), \(D=\mathbb{R}^{n}\backslash \{0\}\) (\(n\geq 3\)) and p is a positive continuous function in D, which may be singular at \(x=0\). Under sufficient conditions for the weighted function \(p(x)\), 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.
1 Introduction and the main result
Semilinear elliptic partial differential equations of the type
will be considered in \(D=\mathbb{R}^{n}\backslash \{0\}\) (\(n\geq 3\)), where \(\gamma <1\) 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 \(\vert x \vert \rightarrow 0\) and as \(\vert x \vert \rightarrow \infty \). 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, [11–13].
The above equation, subjected to homogeneous Dirichlet boundary conditions, has been intensively studied in the case where \(D=\mathbb{R}^{n}\) (\(n\geq 3\)). In this sense, the existence of entire positive solutions for any \(\gamma <0\), 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 [20], the authors proved the existence and uniqueness of a positive continuous solution to the nonlinear elliptic problem
where Ω is an unbounded domain in \(\mathbb{R}^{n}\) (\(n\geq 3\)), with smooth boundary ∂Ω and \(\varphi :\Omega \times (0,\infty )\rightarrow (0,\infty )\) is continuous and nonincreasing with respect to the second variable, such that for all \(c>0\), \(V(\varphi (\cdot ,c))>0\) and \(\varphi (\cdot ,c)\) belongs to \(K_{n}^{\infty }(\Omega )\), where \(V=(-\Delta )^{-1}\) and \(K_{n}^{\infty }(\Omega )\) is the Kato class (see Definition 2.1).
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 \(0<\gamma <1\) and p is a nonnegative measurable function such that the function \(x\rightarrow \int _{\mathbb{R}^{n}} \frac{p(y)}{ \vert x-y \vert ^{n-2}}\,dy \) belongs to \(L^{\infty }(\mathbb{R}^{n}) \).
In [5], by using Karamata regular variation theory and the sub-supersolutions method, the authors studied the asymptotic behavior as \(\vert x \vert \rightarrow \infty \) of the unique classical positive solution of problem (1.3) with \(\gamma <1\) and \(p(x)\) is a nonnegative function in \(C_{\mathrm{loc}}^{\alpha }(\mathbb{R}^{n})\), \(0<\alpha <1\), such that there exists \(c>0\) satisfying
where \(\lambda \geq 2\) 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 \(\gamma <1\), \(D=\mathbb{R}^{n}\backslash \{0\}\) (\(n\geq 3\)) 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]:
Definition 1.1
A positive continuously differentiable function L defined on \([A,\infty )\), for some \(A>0\) is said to be normalized slowly varying (at infinity) if,
we write \(L\in \mathcal{NSV}_{\infty }\).
As examples, we quote:
-
\(L(t)=\prod^{m}_{k=1}(\ln _{k}t)^{\xi _{k}}\), where \(\ln _{k}t=\ln \ln _{k-1}t\) and \(\xi _{k}\in \mathbb{R}\).
-
\(L(t)=\exp (\prod^{m}_{k=1}(\ln _{k}t)^{\nu _{k}})\), where \(0<\nu _{k}<1\).
-
\(L(t)=\exp \{(\ln t)^{\frac{1}{3}}\cos (\ln t)^{\frac{1}{3}}\}\).
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 \(t\rightarrow \infty \) is limited in the sense that it satisfies for any \(\varepsilon >0\)
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 \(L\in \mathcal{NSV}_{0}\).
Remark 1.3
Note that L belongs to \(\mathcal{NSV}_{0}\) if and only if \(t\rightarrow L(1/t)\) belongs to \(\mathcal{NSV}_{\infty }\).
Throughout this paper, we make the following assumption:
-
(H)
p is a positive continuous function in D such that there exists \(c>0\) satisfying
$$ \frac{1}{c}\mathcal{P}(x)\leq p(x)\leq c\mathcal{P}(x ), \quad \text{for }x\in D, $$(1.7)
where \(\mathcal{P}(x):= \vert x \vert ^{-\mu }\mathcal{L}_{0}(\min ( \vert x \vert ,1))( \vert x \vert +1)^{\mu -\lambda }\mathcal{L}_{\infty }(\max ( \vert x \vert ,1))\), with \(\gamma <1\), \(\mu \leq n+(2-n)\gamma \) and \(\lambda \geq 2\).
Here, \(\mathcal{L}_{0}\in \mathcal{NSV}_{0}\), defined on \((0,a]\), for some \(a>1\) and \(\mathcal{L}_{\infty }\in \mathcal{NSV}_{\infty }\), defined on \([1,\infty )\) such that
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.
Theorem 1.4
Under assumption (H), problem (1.5) has at least one positive continuous solution u on D such that
where c is a positive constant and for \(x\in D\),
where \(\xi =\min (0,\frac{2-\mu }{1-\gamma })\), \(\zeta =\max (2-n,\frac{2-\lambda }{1-\gamma })\) and \(\widetilde{\mathcal{L}}_{0}\in \mathcal{NSV}_{0}\) (resp., \(\widetilde{\mathcal{L}}_{\infty }\in \mathcal{NSV}_{\infty }\)) is defined on \((0,a)\) (resp., on \([1,\infty )\)) by
and
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:
-
(i)
\(D=\mathbb{R}^{n}\backslash \{0\}\) (\(n\geq 3\)).
-
(ii)
\(\mathcal{B}(D)\) denotes the set of all Borel measurable functions in D and \(\mathcal{B}^{+} ( D ) \) denotes the set of nonnegative ones.
-
(iii)
\(C(D)\) refers to all continuous functions in D.
-
(iv)
\(s\wedge t=\min (s,t)\) and \(s\vee t=\max (s,t)\), for all \(s,t\in \mathbb{R}\).
-
(v)
For \(f,g\in \mathcal{B}^{+} ( D ) \), \(f\approx g\) in D, means that there exists \(c>0\) such that \(\frac{1}{c}f(x)\leq g(x)\leq cf(x)\), for all \(x\in D\).
-
(vi)
\(\mathcal{S}^{+} ( \Omega ) \) denotes the set of all nonnegative superharmonic functions on an open set Ω of \(\mathbb{R}^{n}\).
-
(vii)
For \(x\in D\),
$$ \varrho _{0}(x):= \frac{1+ \vert x \vert ^{n-2}}{ \vert x \vert ^{n-2}}. $$Note that \(\varrho _{0}\in \mathcal{S}^{+} ( \mathbb{R}^{n} ) \) and harmonic on D, see, for example, [3].
-
(viii)
For \(x,y\in \mathbb{R}^{n}\), we denote the normalized fundamental solution of Laplace’s equation by:
$$ \Gamma (x,y)=\frac{c_{n}}{ \vert x-y \vert ^{n-2}},\quad \text{with }c_{n}= \frac{\Gamma (\frac{n}{2}-1)}{4\pi ^{\frac{n}{2}}}. $$(1.13) -
(ix)
The Newtonian potential \(\mathcal{N}\) is defined on \(\mathcal{B}^{+} ( D ) \) by
$$ \mathcal{N}f(x)= \int _{D}\Gamma (x,y)f(y)\,dy . $$(1.14)
From [6, Proposition 2.10], we learned that if \(f\in \mathcal{B}^{+} ( D ) \) such that \(f\in L_{\mathrm{loc}}^{1}(D)\) and \(\mathcal{N}f\in L_{\mathrm{loc}}^{1}(D)\), then
Throughout this paper, the letter c will denote a generic positive constant that may vary from line to line.
2 Preliminaries
2.1 Kato class \(K_{n}^{\infty }(D)\)
Definition 2.1
(See [28])
A function ψ in \(\mathcal{B}(D)\) is said to be in the Kato class \(K_{n}^{\infty }(D)\) if
and
where \(\Gamma (x,y)\) is given by (1.13).
Example 2.2
Let \(p>\frac{n}{2}\). Then, we have
Indeed, for \(\psi \in L^{p} ( D ) \), by using the Hölder inequality, it is clear that (2.1) holds. Now, assume further that \(\psi \in L^{1} ( D ) \), then
Hence, ψ satisfies (2.2).
The next Lemma is due to Mâagli and Zribi, see [20, Remark 2 and Proposition 1].
Lemma 2.3
-
(i)
Let ψ be a radial function in D, then
$$ \psi \in K_{n}^{\infty }(D)\quad \textit{if and only if}\quad \int _{0}^{\infty }r \bigl\vert \psi (r) \bigr\vert \,dr< \infty . $$ -
(ii)
Let \(\psi \in \mathcal{B}(D)\) satisfying (2.1). Then, for each \(M>0\), we have
$$ \int _{D\cap ( \vert y \vert \leq M)} \bigl\vert \psi (y) \bigr\vert \,dy < \infty . $$
Remark 2.4
For all \(x,y,z\in \mathbb{R}^{n}\), we have
where \(c_{n}=\frac{\Gamma (\frac{n}{2}-1)}{4\pi ^{\frac{n}{2}}}\).
Proposition 2.5
Let \(\psi \in K_{n}^{\infty }(D)\), \(x_{0}\in \mathbb{R}^{n}\) and \(h\in \mathcal{S}^{+} ( D ) \). Then, we have
and
Proof
Since \(h\in \mathcal{S}^{+} ( D ) \), then by [23, Theorem 2.1, p. 164], there exists a sequence \((h_{k})_{k}\subset \mathcal{B}^{+}(D)\) such that
Therefore, we need to prove (2.4) and (2.5) only for \(h(y)=\Gamma (y,z)\) uniformly in \(z\in D\).
Let \(r>0\). By using Remark 2.4, there exists a constant \(c>0\), such that for all \(x,y,z\in D\),
For \(\varepsilon >0\), 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 \(r\rightarrow 0\).
Finally, note that assertion (2.5) follows by using similar arguments as above. □
Proposition 2.6
Let \(\psi \in K_{n}^{\infty }(D)\) and \(\varrho _{0}(x):= \frac{1+ \vert x \vert ^{n-2}}{ \vert x \vert ^{n-2}}\). Then, the function
is continuous on \(\mathbb{R}^{n}\) with \(\lim _{ \vert x \vert \rightarrow \infty }v(x)=0\). That is, \(v(x)\in C_{0}(\mathbb{R}^{n})\).
Proof
Let \(\psi \in K_{n}^{\infty }(D)\) and \(x_{0}\in \mathbb{R}^{n}\). Since \(\varrho _{0}\in \mathcal{S}^{+} ( D ) \), then for \(\varepsilon >0\), by Proposition 2.5, there exists \(M>r>0\), such that the following holds:
(i) If \(x_{0}\neq 0\), then for \(x\in B(x_{0},\frac{r}{2})\cap D\), we have
where \(D_{0}=D\cap B^{c}(0,r)\cap B^{c}(x_{0},r)\).
Since \((x,y)\mapsto \frac{1}{\varrho _{0}(x)}\Gamma (x,y)\) is continuous on \(( B(x_{0},\frac{r}{2})\cap D ) \times (D_{0}\cap ( \vert y \vert \leq M))\), we obtain by Lemma 2.3 (ii) and Lebesgue’s dominated convergence theorem,
Hence, there exists \(\delta >0\) with \(\delta <\frac{r}{2}\) such that if \(x\in B(x_{0},\delta )\cap D\),
Hence, for \(x\in B(x_{0},\delta )\cap D\), we have
That is,
(ii) If \(x_{0}=0\) and \(x\in B(0,\frac{r}{2})\cap D\), then we have
Now, since \(\lim _{ \vert x \vert \rightarrow 0}\frac{1}{\varrho _{0}(x)}\Gamma (x,y)\varrho _{0}(y)=0\), for all \(y\in D\cap B^{c}(0,r)\cap ( \vert y \vert \leq M)\), we deduce by similar arguments as above that
(iii) It remains to prove that \(\lim _{ \vert x \vert \rightarrow \infty }v(x)=0\).
To this end, let \(x\in D\) such that \(\vert x \vert \geq M+1\). Using Proposition 2.5 and Lemma 2.3 (ii), we deduce that
where c is some positive constant.
This implies that \(\lim _{ \vert x \vert \rightarrow \infty }v(x)=0\). □
2.2 Karamata regular variation theory
Let us recall some basic properties of Karamata regular variation theory (see [2, 14, 21, 24, 25]).
The following result concerns operations that preserve slow variation.
Proposition 2.7
If \(L_{1}(t)\), \(L_{2}(t)\) are slowly varying at infinity (resp., at zero), then the same holds for \(L_{1}(t)+L_{2}(t)\), \(L_{1}(t)L_{2}(t)\), and \((L_{1}(t))^{\nu }\) for any \(\nu \in \mathbb{R}\).
Proposition 2.8
-
(i)
If \(L(t)\in \mathcal{NSV}_{\infty }\), then for any \(\varepsilon >0\)
$$ \lim _{t\rightarrow \infty }t^{\varepsilon }L(t)=\infty ,\qquad \lim _{t\rightarrow \infty }t^{-\varepsilon }L(t)=0. $$ -
(ii)
If \(L(t)\in \mathcal{NSV}_{0}\), then for any \(\varepsilon >0 \)
$$ \lim _{t\rightarrow 0^{+}}t^{\varepsilon }L(t)=0 \quad \textit{and}\quad \lim_{t\rightarrow 0^{+}} t^{-\varepsilon }L(t)=\infty . $$
The following result, termed Karamata’s integration theorem, will be used later.
Proposition 2.9
Let \(L(t)\in \mathcal{NSV}_{\infty }\). Then,
-
(i)
if \(\nu >-1\),
$$ \int _{A}^{t}s^{\nu }L(s)\,ds\sim \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow \infty ; $$ -
(ii)
if \(\nu <-1\),
$$ \int _{t}^{\infty }s^{\nu }L(s)\,ds\sim - \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow \infty ; $$ -
(iii)
if \(\nu =-1\),
$$ l(t)= \int _{A}^{t}s^{-1}L(s)\,ds\in \mathcal{NSV}_{\infty }\quad \textit{and}\quad \lim _{t\rightarrow \infty } \frac{L(t)}{l(t)}=0; $$
and if \(\int _{A}^{\infty }s^{-1}L(s)\,ds<\infty \),
The following is an analog of Proposition 2.9 for L defined at zero instead of ∞.
Proposition 2.10
Let \(L(t)\in \mathcal{NSV}_{0}\). Then,
-
(i)
if \(\nu >-1\),
$$ \int _{0}^{t}s^{\nu }L(s)\,ds\sim \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow 0^{+}; $$ -
(ii)
if \(\nu <-1\),
$$ \int _{t}^{a}s^{\nu }L(s)\,ds\sim - \frac{1}{\nu +1}t^{\nu +1}L(t),\quad t\rightarrow 0^{+}; $$ -
(iii)
if \(\nu =-1\),
$$ l_{0}(t)= \int _{t}^{a}s^{-1}L(s)\,ds\in \mathcal{NSV}_{0}\quad \textit{and} \quad \lim _{t\rightarrow 0^{+}} \frac{L(t)}{l_{0}(t)}=0; $$
and if \(\int _{0}^{a}s^{-1}L(s)\,ds<\infty \),
The following result, will play a central role in establishing our main result in Sect. 3.
Proposition 2.11
For \(\alpha \leq n\) and \(\beta \geq 2\), set
where \(L_{0}\in \mathcal{NSV}_{0}\) defined on \((0,a]\), for some \(a>1\) and \(L_{\infty }\in \mathcal{NSV}_{\infty }\), defined on \([1,\infty )\) such that
Then,
where for \(t\in (0,a)\),
and for \(t\geq 1\),
Proof
Since b is a nonnegative radial measurable function on D, it follows from [23, Proposition 1.7], that
where the function J is defined on \([0,\infty )\) by
We need to estimate \(J(t)\). Note that, under condition (2.7), \(J(t)<\infty \).
Let \(a>1\), then we have
We discuss the following cases:
Case 1. \(0< t\leq 1\). Clearly from (2.7), we have
On the other hand, by writing
we deduce that
Therefore, by (2.7) and Propositions 2.8 and 2.10, we obtain
That is,
Case 2. \(t\geq a+1\). From (2.7), we derive that
On the other hand, since
we deduce that
Hence, by (2.7) and Propositions 2.8 and 2.9, we obtain
Therefore, by using Proposition 2.9 and [5, Lemma 2.3], we conclude that
Hence,
Finally, since \(J(t)\), \(\phi _{0}(t)\) and \(\phi _{\infty }(t)\) are positive continuous functions on \([1,a+1]\), we deduce that
Hence, by combining (2.8), (2.9) and (2.10), we obtain
This completes the proof. □
Proposition 2.12
Assume that p satisfies hypothesis (H), then
where \(\gamma <1\) and θ is given in (1.10).
Proof
Using (1.7) and (1.10), we obtain
where \(\alpha :=\mu -\gamma \min (0,\frac{2-\mu }{1-\gamma })\) and \(\beta :=\lambda -\gamma \max (2-n,\frac{2-\lambda }{1-\gamma })\).
From the fact that \(\mu \leq n+(2-n)\gamma \) and \(\lambda \geq 2\), we derive that \(\alpha \leq n\) and \(\beta \geq 2\).
By using the basic properties of Karamata regular variation theory and Proposition 2.11 with \(L_{0}=\mathcal{L}_{0}( \vert x \vert \wedge 1) ( \widetilde{\mathcal{L}}_{0}(\min ( \vert x \vert ,1)) ) ^{\frac{\gamma }{1-\gamma }}\in \mathcal{NSV}_{0}\) and \(L_{\infty }=\mathcal{L}_{\infty }( \vert x \vert \vee 1) ( \widetilde{\mathcal{L}}_{\infty }( \vert x \vert \vee 1) ) ^{\frac{\gamma }{1-\gamma }}\in \mathcal{NSV}_{\infty }\), we deduce that
Since \(\min (0,2-\alpha )=\min (0,\frac{2-\mu }{1-\gamma }):=\xi \) and \(\max (2-n,2-\beta )=\max (2-n,\frac{2-\lambda }{1-\gamma }):=\zeta \), we deduce that
This completes the proof. □
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 \(\nu >0\), we denote by \(( P_{\nu } ) \) the following problem
We recall that for \(x\in D\), \(\varrho _{0}(x)= \frac{1+ \vert x \vert ^{n-2}}{ \vert x \vert ^{n-2}}\).
Note that \(\nu \varrho _{0}(x)\) is a solution of the following homogeneous problem
Problem \(( P_{\nu } ) \) can be seen as a perturbation of problem \((H_{\nu })\).
Proposition 3.1
Let \(\gamma <0\) and assume that hypothesis (H) is satisfied. Then, for each \(\nu >0\), problem \(( P_{\nu } ) \) has at least one positive solution \(u_{\nu }\in C(D\cup \{\infty \})\) satisfying for \(x\in D\)
In particular,
Proof
Let \(\gamma <0\) and \(\nu >0\). Due to Lemma 2.3 (i) and hypothesis \((H)\), the function \(\psi (y):=(\varrho _{0}(y))^{(\gamma -1)}p(y)\) becomes in \(K_{n}^{\infty }(D)\).
Therefore, by Proposition 2.6, we have
Let \(\beta_{0} :=\nu +\nu ^{\gamma } \Vert h \Vert _{\infty }\) and consider the convex set Λ given by
Define the operator T on Λ by
Since for all \(\vartheta \in \Lambda \), \(\vartheta ^{\gamma }\leq \nu ^{\gamma }\), then as in the proof of Proposition 2.6 we show that the family TΛ is equicontinuous in \(\mathbb{R}^{n}\cup \{\infty \}\). In particular, for all \(\vartheta \in \Lambda \), \(T\vartheta \in C ( \mathbb{R}^{n}\cup \{\infty \} ) \) and so \(T\Lambda \subset \Lambda \).
Moreover, the family \(\{T\vartheta (x),\vartheta \in \Lambda \}\) is uniformly bounded in \(\mathbb{R}^{n}\cup \{\infty \}\), then by the Arzela–Ascoli theorem (see, for example [9, p. 62]) the set \(T ( \Lambda ) \) becomes relatively compact in \(C ( \mathbb{R}^{n}\cup \{\infty \} ) \).
To prove the continuity of T in Λ, we consider a sequence \(( \vartheta _{k} ) _{k}\subset \Lambda \) and \(\vartheta \in \Lambda \) such that \(\Vert \vartheta _{k}-\vartheta \Vert _{\infty } \rightarrow 0\) as \(k\rightarrow \infty \). Then, we have
Now, since
we deduce by the dominated convergence theorem and Proposition 2.6 that
Since \(T ( \Lambda ) \) is relatively compact in \(C ( \mathbb{R}^{n}\cup \{\infty \} ) \), we obtain
Hence, T is a compact mapping of Λ to itself and by the Schauder fixed-point theorem, there exists \(\vartheta _{\nu }\in \Lambda \) such that for each \(x\in \mathbb{R}^{n}\)
Since \(\vartheta _{\nu }^{\gamma }\leq \nu ^{\gamma }\), we deduce from (3.3) and (3.2) that
Put \(u_{\nu }(x)=\varrho _{0}(x)\vartheta _{\nu }(x)\), for \(x\in D \). Then, \(u_{\nu }\in C ( D\cup \{\infty \} ) \) and we have
and
Now, since the function \(y\mapsto p(y)u_{\nu }^{\gamma }(y)\in L_{\mathrm{loc}}^{1}(D)\) and from (3.5) the function \(x\mapsto \mathcal{N}(pu_{\nu }^{\gamma })(x)\in L_{\mathrm{loc}}^{1}(D)\), we deduce by (1.15) that \(u_{\nu }\) satisfies
By (3.4), we have
This completes the proof. □
The next result is due to Mâagli and Zribi, see [20, Lemma 1].
Lemma 3.2
Let g∈ \(\mathcal{B}^{+}(D)\) and \(v\in \mathcal{S}^{+}(D)\). Then, for any w∈ \(\mathcal{B}(D)\) such that \(\mathcal{N} ( g \vert w \vert ) <\infty \) and \(w+\mathcal{N} ( gw ) =v\), we have
Corollary 3.3
Let \(\gamma <0\), \(0<\nu _{1}\leq \nu _{2}\) and \(u_{\nu _{i}}\in C(D\cup \{\infty \})\) be the solution of problem \(( P_{\nu _{i}} ) \) given by (3.1). Then, we have
Proof
Let g be the function defined on D by
Since \(\gamma <0\), then \(g\in \mathcal{B}^{+} ( D ) \) and by (3.1) we have
Using (3.6) and (3.2), we obtain for \(x\in D\),
Hence, by (3.8) and Lemma 3.2 with \(w=u_{\nu _{2}}-u_{{\nu _{1}}}\), we obtain (3.7). □
Proposition 3.4
Let \(\gamma <0\). Under hypothesis (H), problem (1.5) has at least one positive solution \(u\in C(D)\) satisfying for \(x\in D\)
Proof
Let \(( \nu _{k} ) _{k}\) be a positive sequence decreasing to zero. Let \(u_{k }\in C(D\cup \{\infty \})\) be the solution of problem \(( P_{\nu _{k}} ) \) given by (3.1). By Corollary 3.3, the sequence \(( u_{k} ) _{k}\) decreases to a function u, and since \(\gamma <0\) the sequence \(( u_{k}-\nu _{k}\varrho _{0}(x) ) _{k}\) increases to u. Therefore, by using (3.1), (3.6) and the fact that \(\gamma <0\), we obtain for each \(x\in D\),
where \(\beta _{k}:=\nu _{k}+\nu _{k}^{\gamma } \Vert h \Vert _{ \infty }\) and h is given by (3.2).
By the monotone convergence theorem, we obtain
Since for each \(x\in D\), \(u(x)=\inf _{k}u_{k}(x)=\sup _{k} ( u_{k}(x)-\nu _{k}\varrho _{0}(x) )\), then u is an upper and lower semicontinuous function on D and so \(u\in C(D)\).
Since the function \(y\mapsto p(y)u^{\gamma }(y)\) is in \(L_{\mathrm{loc}}^{1}(D)\) and from (3.9) the function \(x\mapsto \mathcal{N}(pu^{\gamma })(x)\) is also in \(L_{\mathrm{loc}}^{1}(D)\), we deduce by (1.15) that
Finally, using the fact that \(u_{k}\) is a solution of problem \(( P_{\nu _{k}} ) \) and that \(0< u(x)\leq u_{k}(x)\), for \(x\in D\), we obtain
Hence, u is a solution of problem (1.5). □
Proof of Theorem 1.4
Under assumption (H), by Proposition 2.12, there exists \(M\geq 1\) such that for each D,
where θ is the function defined in (1.10) and \(q(y):=p(y)\theta ^{\gamma }(y)\).
We split the proof into two cases.
Case 1: \(\gamma <0\).
By Proposition 3.4 problem (1.5) has a positive continuous solution u satisfying (3.9). We claim that u satisfies (1.9).
By (3.10), we have
Let \(m=M^{-\frac{\gamma }{1-\gamma }}\). Then, by elementary calculus we have
where \(f(x):=mp(x) [ \theta ^{\gamma }(x)-M^{\gamma } ( \mathcal{N}q ) ^{\gamma }(x) ] \), for \(x\in D\).
Clearly, we have \(f\in \mathcal{B}^{+} ( D ) \) and by using (3.9) and (3.12), we obtain
Let g be the function defined on D by
Since \(\gamma <0\), then \(g\in \mathcal{B}^{+} ( D ) \) and we have
Therefore, the relation (3.13) becomes
Now, since \(f\in \mathcal{B}^{+} ( D ) \) by using (3.14), (3.9), (3.12) and (3.10), we obtain
Hence, by Lemma 3.2, we obtain
Similarly, we prove that
Thus, by (3.10) u satisfies (1.9).
Case 2: \(0\leq \gamma <1\).
Let \(\omega (x)=\frac{1}{\varrho _{0}(x)}\theta (x)\), for \(x\in D\). By (3.10), we have
Put \(c=M^{\frac{1}{1-\gamma }}\) and consider the closed convex set given by
Note that \(\omega \in E\). So \(E\neq \phi \).
Define the operator \(\mathbb{T}\) on E by
By using (3.15), we obtain for all \(v\in E\),
For all \(v\in E\), we have
Therefore, as in the proof of Proposition 2.6, we deduce that
Hence, \(\mathbb{T} ( E ) \subset E\).
Let \(( \omega _{k} ) _{k}\subset C_{0}(\mathbb{R}^{n})\) be defined by
Since the operator \(\mathbb{T}\) is nondecreasing and \(\mathbb{T} ( E ) \subset E\), we obtain
So, by the convergence monotone theorem, the sequence \(( \omega _{k} ) _{k}\) converges to a function v satisfying for each \(x\in D\),
Since v is bounded, we prove by similar arguments as in the proof of Proposition 2.6 that \(v\in C_{0}(\mathbb{R}^{n})\).
Put \(u(x)=\varrho _{0}(x)v(x)\). Then, \(u\in C(D)\) and satisfies the equation
Finally, since the function \(y\mapsto p(y)u^{\gamma }(y)\) is in \(L_{\mathrm{loc}}^{1}(D)\) and from (3.16) the function \(x\mapsto \mathcal{N}(pu^{\gamma })(x)\) is also in \(L_{\mathrm{loc}}^{1}(D)\), we deduce by (1.15) that u is a solution of problem (1.5). The proof of Theorem 1.4 is completed. □
Example 3.5
Let \(\gamma <1\) and \(p\in C ( D ) \), such that
where \(\mu < n+ ( 2-n ) \gamma \) and \(\beta \in \mathbb{R} \). Then, by Theorem 1.4, problem (1.5) has at least one positive solution \(u\in C ( D ) \) satisfying the following estimates:
-
(i)
If \(2<\mu <n+ ( 2-n ) \gamma \), then for \(x\in D\),
$$ u(x)\thickapprox \vert x \vert ^{\frac{2-\mu }{1-\gamma }} \biggl( \log \biggl( \frac{3}{ \vert x \vert \wedge 1}\biggr) \biggr) ^{ \frac{-\beta }{1-\gamma }} \bigl( \log \bigl(3 \vert x \vert \vee 3\bigr) \bigr) ^{ \frac{-1}{1-\gamma }}. $$In particular, \(\lim _{ \vert x \vert \rightarrow 0}u(x)=\infty \).
-
(ii)
If \(\mu =2\) and \(\beta >1\) or \(\mu <2\), then for \(x\in D\),
$$ u(x)\thickapprox \bigl( \log \bigl(3 \vert x \vert \vee 3\bigr) \bigr) ^{\frac{-1}{1-\gamma }}. $$ -
(iii)
If \(\mu =2\) and \(\beta =1\), then for \(x\in D\),
$$ u(x)\thickapprox \biggl( \log _{2}\biggl( \frac{3}{ \vert x \vert \wedge 1}\biggr) \biggr) ^{\frac{1}{1-\gamma }} \bigl( \log \bigl(3 \vert x \vert \vee 3\bigr) \bigr) ^{\frac{-1}{1-\gamma }}. $$ -
(iv)
If \(\mu =2\) and \(\beta <1\), then for \(x\in D\),
$$ u(x)\thickapprox \biggl( \log \biggl( \frac{3}{ \vert x \vert \wedge 1}\biggr) \biggr) ^{\frac{1-\beta }{1-\gamma }} \bigl( \log \bigl(3 \vert x \vert \vee 3\bigr) \bigr) ^{\frac{-1}{1-\gamma }}. $$
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Bachar, I., Aljarallah, E. Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in \(R^{n}\backslash\{0\}\). Bound Value Probl 2022, 4 (2022). https://doi.org/10.1186/s13661-022-01584-3
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DOI: https://doi.org/10.1186/s13661-022-01584-3