Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions
We consider the elliptic problem with nonlinear boundary conditions: $$displaylines{ -Delta u +bu=f(x,u)quadhbox{in }Omega,cr -partial_{u}u=|u|^{q-1}u-g(u)quadhbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^n$. Proving the existence of solutions of this problem relie...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2012-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2012/33/abstr.html |
| _version_ | 1818788030535696384 |
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| author | Junichi Harada Mitsuharu Otani |
| author_facet | Junichi Harada Mitsuharu Otani |
| author_sort | Junichi Harada |
| collection | DOAJ |
| description | We consider the elliptic problem with nonlinear boundary conditions: $$displaylines{ -Delta u +bu=f(x,u)quadhbox{in }Omega,cr -partial_{u}u=|u|^{q-1}u-g(u)quadhbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^n$. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since $L^{q+1}(partialOmega)subset H^1(Omega)$ does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations. |
| first_indexed | 2024-12-18T14:17:11Z |
| format | Article |
| id | doaj.art-e6fbf09c66344a6290b89efa782415d6 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T14:17:11Z |
| publishDate | 2012-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-e6fbf09c66344a6290b89efa782415d62022-12-21T21:04:58ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912012-02-01201233,19Multiple solutions for semilinear elliptic equations with nonlinear boundary conditionsJunichi HaradaMitsuharu OtaniWe consider the elliptic problem with nonlinear boundary conditions: $$displaylines{ -Delta u +bu=f(x,u)quadhbox{in }Omega,cr -partial_{u}u=|u|^{q-1}u-g(u)quadhbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^n$. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since $L^{q+1}(partialOmega)subset H^1(Omega)$ does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations.http://ejde.math.txstate.edu/Volumes/2012/33/abstr.htmlNonlinear boundary conditions |
| spellingShingle | Junichi Harada Mitsuharu Otani Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions Electronic Journal of Differential Equations Nonlinear boundary conditions |
| title | Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions |
| title_full | Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions |
| title_fullStr | Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions |
| title_full_unstemmed | Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions |
| title_short | Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions |
| title_sort | multiple solutions for semilinear elliptic equations with nonlinear boundary conditions |
| topic | Nonlinear boundary conditions |
| url | http://ejde.math.txstate.edu/Volumes/2012/33/abstr.html |
| work_keys_str_mv | AT junichiharada multiplesolutionsforsemilinearellipticequationswithnonlinearboundaryconditions AT mitsuharuotani multiplesolutionsforsemilinearellipticequationswithnonlinearboundaryconditions |