Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 58, pp. 1-19.
Title: Existence of positive solutions for some polyharmonic nonlinear
boundary-value problems
Authors: Habib Maagli (Campus Univ., Tunis, Tunisia)
Faten Toumi (Campus Univ., Tunis, Tunisia)
Malek Zribi (Campus Univ., Tunis, Tunisia)
Abstract:
We present existence results for the polyharmonic nonlinear
elliptic boundary-value problem
$$\displaylines{
(-\Delta )^m u=f(\cdot,u) \quad \hbox{in }B \cr
(\frac{\partial }{\partial \nu })^j u=0\quad
\hbox{on }\partial B, \quad 0\leq j\leq m-1.
}$$
(in the sense of distributions), where $B$ is the unit ball in
$\mathbb{R}^n$ and $n\geq 2$. The nonlinearity $f(x,t)$ satisfies
appropriate conditions related to a Kato class of functions $K_{m,n}$.
Our approach is based on estimates for the polyharmonic Green function
with zero Dirichlet boundary conditions and on the Schauder fixed point
theorem.
Submitted April 2, 2003. Published May 20, 2003.
Math Subject Classifications: 34B27, 35J40
Key Words: Green function; positive solution; Schauder fixed point theorem;
singular polyharmonic elliptic equation.