Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems
In this article, we are concerned with the asymptotic behavior of the classical solution to the semilinear boundary-value problem $$ Delta u+a(x)u^{sigma }=0 $$ in $mathbb{R}^n$, $u>0$, $lim_{|x|o infty }u(x)=0$, where $sigma <1$. The special feature is to consider the function $a$...
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| Format: | Article |
| Language: | English |
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Texas State University
2011-07-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2011/88/abstr.html |
| _version_ | 1818777082551861248 |
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| author | Rym Chemmam Abdelwaheb Dhifli Habib Maagli |
| author_facet | Rym Chemmam Abdelwaheb Dhifli Habib Maagli |
| author_sort | Rym Chemmam |
| collection | DOAJ |
| description | In this article, we are concerned with the asymptotic behavior of the classical solution to the semilinear boundary-value problem $$ Delta u+a(x)u^{sigma }=0 $$ in $mathbb{R}^n$, $u>0$, $lim_{|x|o infty }u(x)=0$, where $sigma <1$. The special feature is to consider the function $a$ in $C_{m loc}^{alpha }(mathbb{R}^n)$, $0<alpha <1$, such that there exists $c>0$ satisfying $$ frac{1}{c}frac{L(|x| +1)}{(1+|x| )^{lambda }} leq a(x)leq cfrac{L(|x| +1)}{(1+|x| )^{lambda }}, $$ where $L(t):=exp ig(int_1^tfrac{z(s)}{s}dsig)$, with $zin C([1,infty ))$ such that $lim_{to infty } z(t)=0$. The comparable asymptotic rate of $a(x)$ determines the asymptotic behavior of the solution. |
| first_indexed | 2024-12-18T11:23:10Z |
| format | Article |
| id | doaj.art-c43aa8b61d474cbfbd7a9cca567678d6 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T11:23:10Z |
| publishDate | 2011-07-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-c43aa8b61d474cbfbd7a9cca567678d62022-12-21T21:09:47ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912011-07-01201188,112Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problemsRym ChemmamAbdelwaheb DhifliHabib MaagliIn this article, we are concerned with the asymptotic behavior of the classical solution to the semilinear boundary-value problem $$ Delta u+a(x)u^{sigma }=0 $$ in $mathbb{R}^n$, $u>0$, $lim_{|x|o infty }u(x)=0$, where $sigma <1$. The special feature is to consider the function $a$ in $C_{m loc}^{alpha }(mathbb{R}^n)$, $0<alpha <1$, such that there exists $c>0$ satisfying $$ frac{1}{c}frac{L(|x| +1)}{(1+|x| )^{lambda }} leq a(x)leq cfrac{L(|x| +1)}{(1+|x| )^{lambda }}, $$ where $L(t):=exp ig(int_1^tfrac{z(s)}{s}dsig)$, with $zin C([1,infty ))$ such that $lim_{to infty } z(t)=0$. The comparable asymptotic rate of $a(x)$ determines the asymptotic behavior of the solution.http://ejde.math.txstate.edu/Volumes/2011/88/abstr.htmlAsymptotic behaviorDirichlet problemground sate solutionsingular equationssublinear equations |
| spellingShingle | Rym Chemmam Abdelwaheb Dhifli Habib Maagli Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems Electronic Journal of Differential Equations Asymptotic behavior Dirichlet problem ground sate solution singular equations sublinear equations |
| title | Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems |
| title_full | Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems |
| title_fullStr | Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems |
| title_full_unstemmed | Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems |
| title_short | Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems |
| title_sort | asymptotic behavior of ground state solutions for sublinear and singular nonlinear dirichlet problems |
| topic | Asymptotic behavior Dirichlet problem ground sate solution singular equations sublinear equations |
| url | http://ejde.math.txstate.edu/Volumes/2011/88/abstr.html |
| work_keys_str_mv | AT rymchemmam asymptoticbehaviorofgroundstatesolutionsforsublinearandsingularnonlineardirichletproblems AT abdelwahebdhifli asymptoticbehaviorofgroundstatesolutionsforsublinearandsingularnonlineardirichletproblems AT habibmaagli asymptoticbehaviorofgroundstatesolutionsforsublinearandsingularnonlineardirichletproblems |