Bifurcation for elliptic forth-order problems with quasilinear source term
We study the bifurcations of the semilinear elliptic forth-order problem with Navier boundary conditions $$\displaylines{ \Delta^2 u - \hbox{div} ( c(x) \nabla u ) = \lambda f(u) \quad \text{in }\Omega, \cr \Delta u = u = 0 \quad\text{on } \partial \Omega. }$$ Where $\Omega \subset \mathbb{R}...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2016-04-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2016/92/abstr.html |
| _version_ | 1818379481229819904 |
|---|---|
| author | Soumaya Saanouni Nihed Trabelsi |
| author_facet | Soumaya Saanouni Nihed Trabelsi |
| author_sort | Soumaya Saanouni |
| collection | DOAJ |
| description | We study the bifurcations of the semilinear elliptic forth-order
problem with Navier boundary conditions
$$\displaylines{
\Delta^2 u - \hbox{div} ( c(x) \nabla u ) = \lambda f(u) \quad
\text{in }\Omega, \cr
\Delta u = u = 0 \quad\text{on } \partial \Omega.
}$$
Where $\Omega \subset \mathbb{R}^n$, $n \geq 2$ is a smooth bounded
domain, f is a positive, increasing and convex source term and
$c(x)$ is a smooth positive function on $\overline{\Omega}$ such
that the $L^\infty$-norm of its gradient is small enough. We prove
the existence, uniqueness and stability of positive solutions. We
also show the existence of critical value $\lambda^*$ and the
uniqueness of its extremal solutions. |
| first_indexed | 2024-12-14T02:03:28Z |
| format | Article |
| id | doaj.art-0891b7bbceb34aebaa728a1994758c50 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-14T02:03:28Z |
| publishDate | 2016-04-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-0891b7bbceb34aebaa728a1994758c502022-12-21T23:20:57ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912016-04-01201692,116Bifurcation for elliptic forth-order problems with quasilinear source termSoumaya Saanouni0Nihed Trabelsi1 Campus Univ., Tunis, Tunisia Higher Institute of Medical Tech., Tunis, Tunisia We study the bifurcations of the semilinear elliptic forth-order problem with Navier boundary conditions $$\displaylines{ \Delta^2 u - \hbox{div} ( c(x) \nabla u ) = \lambda f(u) \quad \text{in }\Omega, \cr \Delta u = u = 0 \quad\text{on } \partial \Omega. }$$ Where $\Omega \subset \mathbb{R}^n$, $n \geq 2$ is a smooth bounded domain, f is a positive, increasing and convex source term and $c(x)$ is a smooth positive function on $\overline{\Omega}$ such that the $L^\infty$-norm of its gradient is small enough. We prove the existence, uniqueness and stability of positive solutions. We also show the existence of critical value $\lambda^*$ and the uniqueness of its extremal solutions.http://ejde.math.txstate.edu/Volumes/2016/92/abstr.htmlBifurcationregularitystabilityquasilinear |
| spellingShingle | Soumaya Saanouni Nihed Trabelsi Bifurcation for elliptic forth-order problems with quasilinear source term Electronic Journal of Differential Equations Bifurcation regularity stability quasilinear |
| title | Bifurcation for elliptic forth-order problems with quasilinear source term |
| title_full | Bifurcation for elliptic forth-order problems with quasilinear source term |
| title_fullStr | Bifurcation for elliptic forth-order problems with quasilinear source term |
| title_full_unstemmed | Bifurcation for elliptic forth-order problems with quasilinear source term |
| title_short | Bifurcation for elliptic forth-order problems with quasilinear source term |
| title_sort | bifurcation for elliptic forth order problems with quasilinear source term |
| topic | Bifurcation regularity stability quasilinear |
| url | http://ejde.math.txstate.edu/Volumes/2016/92/abstr.html |
| work_keys_str_mv | AT soumayasaanouni bifurcationforellipticforthorderproblemswithquasilinearsourceterm AT nihedtrabelsi bifurcationforellipticforthorderproblemswithquasilinearsourceterm |