Electron. J. Diff. Eqns., Vol. 2006(2006), No. 102, pp. 1-9.

Existence of positive solutions for higher order singular sublinear elliptic equations

Imed Bachar

Abstract:
We present existence result for the polyharmonic nonlinear problem
$$\displaylines{ (-\Delta )^{pm} u=\varphi (.,u)+\psi (.,u),\quad \hbox{in }B \cr u greater than 0,\quad \hbox{in }B \cr \lim_{|x|\to 1} \frac{(-\Delta )^{jm}u(x)}{(1-|x|)^{m-1}}=0, \quad 0\leq j\leq p-1, }$$
in the sense of distributions. Here $m,p$ are positive integers, $B$ is the unit ball in $\mathbb{R}^{n}(n\geq 2)$ and the nonlinearity is a sum of a singular and sublinear terms satisfying some appropriate conditions related to a polyharmonic Kato class of functions $\mathcal{J}_{m,n}^{(p)}$.

Submitted May 10, 2006. Published August 31, 2006.
Math Subject Classifications: 34B27, 35J40.
Key Words: Green function; higher-order elliptic equations; positive solution; Schauder fixed point theorem.

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Imed Bachar
Département de mathématiques
Faculté des sciences de Tunis
campus universitaire, 2092 Tunis, Tunisia
email: Imed.Bachar@ipeit.rnu.tn

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