Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain
In this paper, we consider the solutions of the boundary blow-up problem $ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-04-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2022607?viewType=HTML |
| _version_ | 1819060485955256320 |
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| author | Keqiang Li Shangjiu Wang Shaoyong Li |
| author_facet | Keqiang Li Shangjiu Wang Shaoyong Li |
| author_sort | Keqiang Li |
| collection | DOAJ |
| description | In this paper, we consider the solutions of the boundary blow-up problem
$ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $
where $ \gamma > 0, \ \Omega $ is a bounded convex smooth domain and symmetric w.r.t. a direction. $ f $ is a locally Lipschitz continuous and non-decreasing function. We prove symmetry and monotonicity of solutions of the problem above by the moving planes method. A maximum principle in narrow domains plays an important role in proof of the main result. |
| first_indexed | 2024-12-21T14:27:45Z |
| format | Article |
| id | doaj.art-6ef37a5a4ccb4a7ba6406f8147df2d7c |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-21T14:27:45Z |
| publishDate | 2022-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-6ef37a5a4ccb4a7ba6406f8147df2d7c2022-12-21T19:00:35ZengAIMS PressAIMS Mathematics2473-69882022-04-0176108601086610.3934/math.2022607Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domainKeqiang Li0Shangjiu Wang1Shaoyong Li21. College of Digital Technology and Engineering, Ningbo University of Finance and Economics, Ningbo 315175, China2. School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China 3. School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China2. School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, ChinaIn this paper, we consider the solutions of the boundary blow-up problem $ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $ where $ \gamma > 0, \ \Omega $ is a bounded convex smooth domain and symmetric w.r.t. a direction. $ f $ is a locally Lipschitz continuous and non-decreasing function. We prove symmetry and monotonicity of solutions of the problem above by the moving planes method. A maximum principle in narrow domains plays an important role in proof of the main result.https://www.aimspress.com/article/doi/10.3934/math.2022607?viewType=HTMLsemilinear elliptic systemsmoving plane methodmaximum principlesymmetry of large solutions |
| spellingShingle | Keqiang Li Shangjiu Wang Shaoyong Li Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain AIMS Mathematics semilinear elliptic systems moving plane method maximum principle symmetry of large solutions |
| title | Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain |
| title_full | Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain |
| title_fullStr | Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain |
| title_full_unstemmed | Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain |
| title_short | Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain |
| title_sort | symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain |
| topic | semilinear elliptic systems moving plane method maximum principle symmetry of large solutions |
| url | https://www.aimspress.com/article/doi/10.3934/math.2022607?viewType=HTML |
| work_keys_str_mv | AT keqiangli symmetryoflargesolutionsforsemilinearellipticequationsinasymmetricconvexdomain AT shangjiuwang symmetryoflargesolutionsforsemilinearellipticequationsinasymmetricconvexdomain AT shaoyongli symmetryoflargesolutionsforsemilinearellipticequationsinasymmetricconvexdomain |