Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$
Abstract We consider the following singular semilinear problem { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-01-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-022-01584-3 |
| _version_ | 1798026186701406208 |
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| author | Imed Bachar Entesar Aljarallah |
| author_facet | Imed Bachar Entesar Aljarallah |
| author_sort | Imed Bachar |
| collection | DOAJ |
| description | Abstract We consider the following singular semilinear problem { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , $$ \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} $$ where γ < 1 $\gamma <1$ , D = R n ∖ { 0 } $D=\mathbb{R}^{n}\backslash \{0\}$ ( n ≥ 3 $n\geq 3$ ) and p is a positive continuous function in D, which may be singular at x = 0 $x=0$ . Under sufficient conditions for the weighted function p ( x ) $p(x)$ , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained. |
| first_indexed | 2024-04-11T18:31:20Z |
| format | Article |
| id | doaj.art-c48e2ca13e564368b03be2ac61710a2a |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-04-11T18:31:20Z |
| publishDate | 2022-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-c48e2ca13e564368b03be2ac61710a2a2022-12-22T04:09:26ZengSpringerOpenBoundary Value Problems1687-27702022-01-012022111910.1186/s13661-022-01584-3Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$Imed Bachar0Entesar Aljarallah1College of Science, Mathematics Department, King Saud UniversityCollege of Science, Mathematics Department, King Saud UniversityAbstract We consider the following singular semilinear problem { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , $$ \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} $$ where γ < 1 $\gamma <1$ , D = R n ∖ { 0 } $D=\mathbb{R}^{n}\backslash \{0\}$ ( n ≥ 3 $n\geq 3$ ) and p is a positive continuous function in D, which may be singular at x = 0 $x=0$ . Under sufficient conditions for the weighted function p ( x ) $p(x)$ , 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.https://doi.org/10.1186/s13661-022-01584-3Slowly varying functionsKato classAsymptotic propertiesBlow-up |
| spellingShingle | Imed Bachar Entesar Aljarallah Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$ Boundary Value Problems Slowly varying functions Kato class Asymptotic properties Blow-up |
| title | Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$ |
| title_full | Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$ |
| title_fullStr | Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$ |
| title_full_unstemmed | Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$ |
| title_short | Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$ |
| title_sort | existence and asymptotic properties of singular solutions of nonlinear elliptic equations in r n 0 r n backslash 0 |
| topic | Slowly varying functions Kato class Asymptotic properties Blow-up |
| url | https://doi.org/10.1186/s13661-022-01584-3 |
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