Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 19, pp. 1-12.
Title: Nonlinear singular Navier problem of fourth order
Authors: Syrine Masmoudi (Faculte des Sciences de Tunis, Tunisia)
Malek Zribi (Faculte des Sciences de Tunis, Tunisia)
Abstract:
We present an existence result for a nonlinear singular differential
equation of fourth order with Navier boundary conditions.
Under appropriate conditions on the nonlinearity $f(t,x,y)$,
we prove that the problem
$$\displaylines{
L^{2}u=L(Lu) =f(.,u,Lu)\quad \hbox{a.e. in }(0,1), \cr
u'(0) =0,\quad (Lu) '(0)=0,\quad u(1) =0,\quad Lu(1) =0.
}$$
has a positive solution behaving like $(1-t)$ on $[0,1]$.
Here $L$ is a differential operator of second order,
$Lu=\frac{1}{A}(Au')'$. For $f(t,x,y)=f(t,x)$, we prove a
uniqueness result. Our approach is based on estimates for
Green functions and on Schauder's fixed point theorem.
Submitted September 29, 2002. Published February 28, 2003.
Math Subject Classifications: 34B15, 34B27.
Key Words: Nonlinear singular Navier problem; Green function;
positive solution.