Electron. J. Diff. Equ., Vol. 2016 (2016), No. 92, pp. 1-16.

Bifurcation for elliptic forth-order problems with quasilinear source term

Soumaya Saanouni, Nihed Trabelsi

Abstract:
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.

Submitted December 3 2015. Published April 6, 2016.
Math Subject Classifications: 35B32, 35B65, 35B35, 35J62.
Key Words: Bifurcation; regularity; stability; quasilinear.

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Soumaya Sâanouni
University of Tunis El Manar
Faculty of Sciences of Tunis
Department of Mathematics
Campus University 2092 Tunis, Tunisia
email: saanouni.soumaya@yahoo.com
Nihed Trabelsi
University of Tunis El Manar
Higher Institute of Medical Technologies of Tunis
9 Street Dr. Zouhair Essafi 1006 Tunis, Tunisia
email: nihed.trabelsi78@gmail.com

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