Existence of positive solutions of elliptic equations with Hardy term
This paper is devoted to studying the existence of positive solutions of the problem: \begin{equation} \begin{cases}\label{0.1}\tag{$\ast$} -\Delta u=\frac{u^{p}}{|x|^{a}}+h(x,u,\nabla u), & \mbox{in} \ \Omega,\\ u=0, & \mbox{on}\ \partial\Omega,\\ \end{cases} \end{equation} where $\Omega\s...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10449 |
| _version_ | 1797352371338084352 |
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| author | Huimin Yan Junhui Xie |
| author_facet | Huimin Yan Junhui Xie |
| author_sort | Huimin Yan |
| collection | DOAJ |
| description | This paper is devoted to studying the existence of positive solutions of the problem:
\begin{equation}
\begin{cases}\label{0.1}\tag{$\ast$}
-\Delta u=\frac{u^{p}}{|x|^{a}}+h(x,u,\nabla u), & \mbox{in} \ \Omega,\\
u=0, & \mbox{on}\ \partial\Omega,\\
\end{cases}
\end{equation}
where $\Omega\subset \mathbb{R}^{N}(N\geq3)$ is an open bounded smooth domain with boundary $\partial\Omega$, and $1<p<\frac{N-a}{N-2}$, $0<a<2$. Under suitable conditions of $h(x,u,\nabla u)$, we get a priori estimates for the positive solutions of problem \eqref{0.1}. By making use of these estimates and topological degree theory, we further obtain some existence results for the positive solutions of problem \eqref{0.1} when $1<p<\frac{N-a}{N-2}$. |
| first_indexed | 2024-03-08T13:15:29Z |
| format | Article |
| id | doaj.art-9195a07641fc4827a06634f2ce63a2f7 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-03-08T13:15:29Z |
| publishDate | 2024-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-9195a07641fc4827a06634f2ce63a2f72024-01-18T08:29:34ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-01-012024111410.14232/ejqtde.2024.1.110449Existence of positive solutions of elliptic equations with Hardy termHuimin Yan0Junhui Xie1School of Mathematics and Statistics, Hubei Minzu University, Enshi City, ChinaSchool of Mathematics and Statistics, Hubei Minzu University, Enshi City, ChinaThis paper is devoted to studying the existence of positive solutions of the problem: \begin{equation} \begin{cases}\label{0.1}\tag{$\ast$} -\Delta u=\frac{u^{p}}{|x|^{a}}+h(x,u,\nabla u), & \mbox{in} \ \Omega,\\ u=0, & \mbox{on}\ \partial\Omega,\\ \end{cases} \end{equation} where $\Omega\subset \mathbb{R}^{N}(N\geq3)$ is an open bounded smooth domain with boundary $\partial\Omega$, and $1<p<\frac{N-a}{N-2}$, $0<a<2$. Under suitable conditions of $h(x,u,\nabla u)$, we get a priori estimates for the positive solutions of problem \eqref{0.1}. By making use of these estimates and topological degree theory, we further obtain some existence results for the positive solutions of problem \eqref{0.1} when $1<p<\frac{N-a}{N-2}$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10449a priori estimateshardy termpositive solutions |
| spellingShingle | Huimin Yan Junhui Xie Existence of positive solutions of elliptic equations with Hardy term Electronic Journal of Qualitative Theory of Differential Equations a priori estimates hardy term positive solutions |
| title | Existence of positive solutions of elliptic equations with Hardy term |
| title_full | Existence of positive solutions of elliptic equations with Hardy term |
| title_fullStr | Existence of positive solutions of elliptic equations with Hardy term |
| title_full_unstemmed | Existence of positive solutions of elliptic equations with Hardy term |
| title_short | Existence of positive solutions of elliptic equations with Hardy term |
| title_sort | existence of positive solutions of elliptic equations with hardy term |
| topic | a priori estimates hardy term positive solutions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10449 |
| work_keys_str_mv | AT huiminyan existenceofpositivesolutionsofellipticequationswithhardyterm AT junhuixie existenceofpositivesolutionsofellipticequationswithhardyterm |