Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection
In this article we analyze the exact boundary behavior of solutions to the singular nonlinear Dirichlet problem $$\displaylines{ -\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0, \; x \in \Omega,\cr u\big|_{\partial \Omega}=0, }$$ where $\Omega$ is a bounded domain with smooth boun...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/19/abstr.html |
| _version_ | 1819181348461477888 |
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| author | Bo Li Zhijun Zhang |
| author_facet | Bo Li Zhijun Zhang |
| author_sort | Bo Li |
| collection | DOAJ |
| description | In this article we analyze the exact boundary behavior of
solutions to the singular nonlinear Dirichlet problem
$$\displaylines{
-\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0, \; x \in \Omega,\cr
u\big|_{\partial \Omega}=0,
}$$
where $\Omega$ is a bounded domain with smooth
boundary in $\mathbb{R}^N$, $q\in (0, 2]$, $\sigma>0$,
$\lambda> 0$, $g\in C^1((0,\infty), (0,\infty))$,
$\lim_{s \to 0^+}g(s)=\infty$, $g$ is decreasing on $(0, s_0)$
for some $s_0>0$, $b \in C_{\rm loc}^{\alpha}({\Omega})$ for some
$\alpha\in (0, 1)$, is positive in $\Omega$, but may be vanishing or
singular on the boundary. We show that $\lambda |\nabla u|^q$
does not affect the first expansion of classical solutions near the
boundary. |
| first_indexed | 2024-12-22T22:28:48Z |
| format | Article |
| id | doaj.art-ebf88ad2765a4bdfb5dd389e78c45af4 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-22T22:28:48Z |
| publishDate | 2015-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-ebf88ad2765a4bdfb5dd389e78c45af42022-12-21T18:10:30ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912015-01-01201519,118Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convectionBo Li0Zhijun Zhang1 Lanzhou Univ., Gansu, China Yantai Univ., Shandong, China In this article we analyze the exact boundary behavior of solutions to the singular nonlinear Dirichlet problem $$\displaylines{ -\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0, \; x \in \Omega,\cr u\big|_{\partial \Omega}=0, }$$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $q\in (0, 2]$, $\sigma>0$, $\lambda> 0$, $g\in C^1((0,\infty), (0,\infty))$, $\lim_{s \to 0^+}g(s)=\infty$, $g$ is decreasing on $(0, s_0)$ for some $s_0>0$, $b \in C_{\rm loc}^{\alpha}({\Omega})$ for some $\alpha\in (0, 1)$, is positive in $\Omega$, but may be vanishing or singular on the boundary. We show that $\lambda |\nabla u|^q$ does not affect the first expansion of classical solutions near the boundary.http://ejde.math.txstate.edu/Volumes/2015/19/abstr.htmlSemilinear elliptic equationsingular Dirichlet problemnonlinear convection term classical solutionboundary behavior |
| spellingShingle | Bo Li Zhijun Zhang Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection Electronic Journal of Differential Equations Semilinear elliptic equation singular Dirichlet problem nonlinear convection term classical solution boundary behavior |
| title | Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection |
| title_full | Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection |
| title_fullStr | Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection |
| title_full_unstemmed | Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection |
| title_short | Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection |
| title_sort | boundary behavior of solutions to a singular dirichlet problem with a nonlinear convection |
| topic | Semilinear elliptic equation singular Dirichlet problem nonlinear convection term classical solution boundary behavior |
| url | http://ejde.math.txstate.edu/Volumes/2015/19/abstr.html |
| work_keys_str_mv | AT boli boundarybehaviorofsolutionstoasingulardirichletproblemwithanonlinearconvection AT zhijunzhang boundarybehaviorofsolutionstoasingulardirichletproblemwithanonlinearconvection |