Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions
We study the nonlinear elliptic problem with discontinuous nonlinearity $$displaylines{ -Delta u = f(u)H(u-mu ) quadhbox{in } Omega, cr u =h quad hbox{on }partial Omega, }$$ where $H$ is the Heaviside unit function, $f,h$ are given functions and $mu$ is a positive real parameter. The domain $O...
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| Format: | Article |
| Language: | English |
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Texas State University
2010-04-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2010/56/abstr.html |
| _version_ | 1831559815322664960 |
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| author | Sabri Bensid Sidi Mohammed Bouguima |
| author_facet | Sabri Bensid Sidi Mohammed Bouguima |
| author_sort | Sabri Bensid |
| collection | DOAJ |
| description | We study the nonlinear elliptic problem with discontinuous nonlinearity $$displaylines{ -Delta u = f(u)H(u-mu ) quadhbox{in } Omega, cr u =h quad hbox{on }partial Omega, }$$ where $H$ is the Heaviside unit function, $f,h$ are given functions and $mu$ is a positive real parameter. The domain $Omega$ is the unit ball in $mathbb{R}^n$ with $ngeq 3$. We show the existence of a positive solution $u$ and a hypersurface separating the region where $-Delta u=0$ from the region where $-Delta u=f(u)$. Our method relies on the implicit function theorem and bifurcation analysis. |
| first_indexed | 2024-12-17T05:34:41Z |
| format | Article |
| id | doaj.art-a8c4b6080186469bbf27393c976d77f8 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-17T05:34:41Z |
| publishDate | 2010-04-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-a8c4b6080186469bbf27393c976d77f82022-12-21T22:01:38ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912010-04-01201056,116Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditionsSabri BensidSidi Mohammed BouguimaWe study the nonlinear elliptic problem with discontinuous nonlinearity $$displaylines{ -Delta u = f(u)H(u-mu ) quadhbox{in } Omega, cr u =h quad hbox{on }partial Omega, }$$ where $H$ is the Heaviside unit function, $f,h$ are given functions and $mu$ is a positive real parameter. The domain $Omega$ is the unit ball in $mathbb{R}^n$ with $ngeq 3$. We show the existence of a positive solution $u$ and a hypersurface separating the region where $-Delta u=0$ from the region where $-Delta u=f(u)$. Our method relies on the implicit function theorem and bifurcation analysis.http://ejde.math.txstate.edu/Volumes/2010/56/abstr.htmlGreen functionmaximum principlebifurcationfree boundary problem |
| spellingShingle | Sabri Bensid Sidi Mohammed Bouguima Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions Electronic Journal of Differential Equations Green function maximum principle bifurcation free boundary problem |
| title | Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions |
| title_full | Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions |
| title_fullStr | Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions |
| title_full_unstemmed | Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions |
| title_short | Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions |
| title_sort | existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions |
| topic | Green function maximum principle bifurcation free boundary problem |
| url | http://ejde.math.txstate.edu/Volumes/2010/56/abstr.html |
| work_keys_str_mv | AT sabribensid existenceandmultiplicityofsolutionstoellipticproblemswithdiscontinuitiesandfreeboundaryconditions AT sidimohammedbouguima existenceandmultiplicityofsolutionstoellipticproblemswithdiscontinuitiesandfreeboundaryconditions |