Electron. J. Diff. Eqns., Vol. 2003(2003), No. 19, pp. 1-12.

Nonlinear singular Navier problem of fourth order

Syrine Masmoudi & Malek Zribi

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.

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Syrine Masmoudi
Departement de Mathematiques
Faculte des Sciences de Tunis
Campus Universitaire, 1060 Tunis, Tunisia
e-amil: Syrine.Sassi@fst.rnu.tn
Malek Zribi
Departement de Mathematiques
Faculte des Sciences de Tunis
Campus Universitaire, 1060 Tunis, Tunisia
e-mail: Malek.Zribi@insat.rnu.tn

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