Nonlinear singular Navier problem of fourth order

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.

Keywords

• Nonlinear singular Navier problem
• Green function
• positive solution.