Makkia Dammak - University of Tunis El Manar, Higher Institute of Medical Technologies of Tunis ۰۹ doctor Zouhair Essafi Street ۱۰۰۶ Tunis,Tunisia
Majdi El Ghord - University of Tunis El Manar, Faculty of Sciences of Tunis, Campus Universities ۲۰۹۲ Tunis, Tunisia

In this paper, we investigate the existence of positive solutions for the ellipticequation \Delta^{۲}\,u+c(x)u = \lambda f(u) on a bounded smooth domain \Omega of \R^{n}, n\geq۲, with Navier boundary conditions. We show that there exists an extremal parameter\lambda^{\ast}>۰ such that for \lambda< \lambda^{\ast}, the above problem has a regular solution butfor \lambda> \lambda^{\ast}, the problem has no solution even in the week sense.We also show that \lambda^{\ast}=\frac{\lambda_{۱}}{a} if \lim_{t\rightarrow \infty}f(t)-at=l\geq۰ and for \lambda< \lambda^{\ast}, the solution is unique but for l<۰ and \frac{\lambda_{۱}}{a}<\lambda< \lambda^{\ast}, the problem has two branches of solutions, where \lambda_{۱} is the first eigenvalue associated to the problem.