Electron. J. Diff. Eqns., Vol. 2003(2003), No. 58, pp. 1-19.

Existence of positive solutions for some polyharmonic nonlinear boundary-value problems

Habib Maagli, Faten Toumi, & Malek Zribi

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.

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Habib Maagli
Departement de Mathematiques
Faculte des Sciences de Tunis
Campus universitaire, 1060 Tunis, Tunisia
email: habib.maagli@fst.rnu.tn
Faten Toumi
Departement de Mathematiques
Faculte des Sciences de Tunis
Campus universitaire, 1060 Tunis, Tunisia
email: toumifaten@yahoo.fr
Malek Zribi
Departement de Mathematiques
Faculte des Sciences de Tunis
Campus universitaire, 1060 Tunis, Tunisia
email: malek.Zribi@insat.rnu.tn

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