Nonexistence results for semilinear systems in unbounded domains
This paper concerns the non-existence of nontrivial solutions for the semi-linear system of gradient type $$displaylines{ lambda frac{partial ^{2}u_{k}}{partial t^{2}} -sum_{i=1}^n frac{partial }{partial x_{i}}(p_{i}(x)frac{ partial u_{k}}{partial x_{i}})+f_{k}(x,u_{1},dots ,u_{m}) =0quad ext{in }O...
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| Format: | Article |
| Language: | English |
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Texas State University
2009-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2009/02/abstr.html |
| _version_ | 1818543636892090368 |
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| author | Abdelkrim Moussaoui Brahim Khodja |
| author_facet | Abdelkrim Moussaoui Brahim Khodja |
| author_sort | Abdelkrim Moussaoui |
| collection | DOAJ |
| description | This paper concerns the non-existence of nontrivial solutions for the semi-linear system of gradient type $$displaylines{ lambda frac{partial ^{2}u_{k}}{partial t^{2}} -sum_{i=1}^n frac{partial }{partial x_{i}}(p_{i}(x)frac{ partial u_{k}}{partial x_{i}})+f_{k}(x,u_{1},dots ,u_{m}) =0quad ext{in }Omega ,; k=1,dots ,m }$$ with Dirichlet, Neumann or Robin boundary conditions. The functions $f_{k}:mathcal{D}imes mathbb{R}^{m}o mathbb{R}$ $(k=1,dots ,m)$ are locally Lipschitz continuous and satisfy $$ 2H(x,u_{1},dots ,u_{m})-sum_{k=1}^m u_{k}f_{k}(x,u_{1},dots ,u_{m})geq 0quad (ext{resp.}leq 0) $$ for $lambda >0$ (resp. $lambda <0$). We establish the non-existence of nontrivial solutions using Pohozaev-type identities. Here $u_{1},dots ,u_{m}$ are in $H^{2}(Omega )cap L^{infty }(Omega )$, $Omega =mathbb{R}imes mathcal{D}$ with $mathcal{D}=prod_{i=1}^n (alpha _{i},eta _{i})$ and $Hin mathcal{C}^{1}(overline{mathcal{D}}imes mathbb{R}^{m})$ such that $frac{partial H}{partial u_{k}}=f_{k}$, $k=1,dots ,m $. |
| first_indexed | 2024-12-11T22:38:00Z |
| format | Article |
| id | doaj.art-b8b88d6c1f0f4096bf0f8ea982e8d0f8 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-11T22:38:00Z |
| publishDate | 2009-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-b8b88d6c1f0f4096bf0f8ea982e8d0f82022-12-22T00:47:54ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912009-01-01200902,111Nonexistence results for semilinear systems in unbounded domainsAbdelkrim MoussaouiBrahim KhodjaThis paper concerns the non-existence of nontrivial solutions for the semi-linear system of gradient type $$displaylines{ lambda frac{partial ^{2}u_{k}}{partial t^{2}} -sum_{i=1}^n frac{partial }{partial x_{i}}(p_{i}(x)frac{ partial u_{k}}{partial x_{i}})+f_{k}(x,u_{1},dots ,u_{m}) =0quad ext{in }Omega ,; k=1,dots ,m }$$ with Dirichlet, Neumann or Robin boundary conditions. The functions $f_{k}:mathcal{D}imes mathbb{R}^{m}o mathbb{R}$ $(k=1,dots ,m)$ are locally Lipschitz continuous and satisfy $$ 2H(x,u_{1},dots ,u_{m})-sum_{k=1}^m u_{k}f_{k}(x,u_{1},dots ,u_{m})geq 0quad (ext{resp.}leq 0) $$ for $lambda >0$ (resp. $lambda <0$). We establish the non-existence of nontrivial solutions using Pohozaev-type identities. Here $u_{1},dots ,u_{m}$ are in $H^{2}(Omega )cap L^{infty }(Omega )$, $Omega =mathbb{R}imes mathcal{D}$ with $mathcal{D}=prod_{i=1}^n (alpha _{i},eta _{i})$ and $Hin mathcal{C}^{1}(overline{mathcal{D}}imes mathbb{R}^{m})$ such that $frac{partial H}{partial u_{k}}=f_{k}$, $k=1,dots ,m $.http://ejde.math.txstate.edu/Volumes/2009/02/abstr.htmlSemi linear systemsPohozaev identitytrivial solutionRobin boundary condition |
| spellingShingle | Abdelkrim Moussaoui Brahim Khodja Nonexistence results for semilinear systems in unbounded domains Electronic Journal of Differential Equations Semi linear systems Pohozaev identity trivial solution Robin boundary condition |
| title | Nonexistence results for semilinear systems in unbounded domains |
| title_full | Nonexistence results for semilinear systems in unbounded domains |
| title_fullStr | Nonexistence results for semilinear systems in unbounded domains |
| title_full_unstemmed | Nonexistence results for semilinear systems in unbounded domains |
| title_short | Nonexistence results for semilinear systems in unbounded domains |
| title_sort | nonexistence results for semilinear systems in unbounded domains |
| topic | Semi linear systems Pohozaev identity trivial solution Robin boundary condition |
| url | http://ejde.math.txstate.edu/Volumes/2009/02/abstr.html |
| work_keys_str_mv | AT abdelkrimmoussaoui nonexistenceresultsforsemilinearsystemsinunboundeddomains AT brahimkhodja nonexistenceresultsforsemilinearsystemsinunboundeddomains |