Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations

In this paper, we investigate the existence of positive solutions for the elliptic equation $\Delta^{2}\,u(x)u = \lambda f(u)$ on a bounded smooth domain $\Omega$ of $\R^{n}$, $n\geq2$, with Navier boundary conditions. We show that there exists an extremal parameter $\lambda^{\ast}>0$ such that for $\lambda \lambda^{\ast}$, the problem has no solution even in the week sense. We also show that $\lambda^{\ast}=\frac{\lambda_{1}}{a}$ if $ \lim_{t\rightarrow \infty}f(t)-at=l\geq0$ and for $\lambda