Dammak, M., El Ghord, M. (2017). Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations. Theory of Approximation and Applications, 11(1), 13-37.

Makkia Dammak; Majdi El Ghord. "Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations". Theory of Approximation and Applications, 11, 1, 2017, 13-37.

Dammak, M., El Ghord, M. (2017). 'Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations', Theory of Approximation and Applications, 11(1), pp. 13-37.

Dammak, M., El Ghord, M. Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations. Theory of Approximation and Applications, 2017; 11(1): 13-37.

Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations

^{1}University of Tunis El Manar, Higher Institute of Medical Technologies of Tunis 09 doctor Zouhair Essafi Street 1006 Tunis,Tunisia

^{2}University of Tunis El Manar, Faculty of Sciences of Tunis, Campus Universities 2092 Tunis, Tunisia

Abstract

In this paper, we investigate the existence of positive solutions for the elliptic equation $\Delta^{2}\,u+c(x)u = \lambda f(u)$ 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 above problem has a regular solution but 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 $\lambda< \lambda^{\ast}$, 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 the first eigenvalue associated to the problem.