Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ 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 $lambda^{ast}=frac{lambda_{1}}{a}$ if$ lim_{trightarrow infty}f(t)-at=lgeq0$ and for $lambda< lambda^{ast}$, the solution is unique but for $l