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.