Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations

Subject Areas : Applied Mathematics

Makkia  Dammak ¹ (University of Tunis El Manar, Higher Institute of Medical Technologies of Tunis 09 doctor Zouhair Essafi Street 1006 Tunis,Tunisia)
Majdi  El Ghord ² (University of Tunis El Manar, Faculty of Sciences of Tunis, Campus Universities 2092 Tunis, Tunisia)

Received: 2015-02-22 Accepted : 2015-07-19 Published : 2017-05-01

Keywords: asymptotically linear, extremal solution, stable minimal solution, regularity,

Abstract :

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 $\R^{n}$, $n\geq2$, with Navierboundary 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_{t\rightarrow \infty}f(t)-at=l\geq0$ and for $\lambda< \lambda^{\ast}$, the solution is unique but for $l<0$ and $\frac{\lambda_{1}}{a}<\lambda< \lambda^{\ast}$, the problem has two branches of solutions, where $\lambda_{1}$ is thefirst eigenvalue associated to the problem.