Abstract:

In this paper, two dimensional singular nonlinear poly harmonic equation of the form Δ((Δ\+nu)\+\{p-1*\}) = f(|x|, u, |u|)u\+\{-β\},\ x∈R\+2 is  considered, where \$p>1, β≥0, n\$ is an integer \$(n≥1),ξ\+\{α*\}:=|ξ|\+\{α-1\}ξ,ξ∈R,α>0.\$   and \$f: [AKR-]\-+×R\-+×[AKR-]\-+→R\-+\$ is a continuous function. It is shown that any positive radially symmet ric entire solution grows at least as fast as positive constant multiples of \$|x|\+\{2n\}(\%log\%|x|)\+\{1/(p-1)\}\$ as \$|x|→∞\$. It is given that some sufficient conditions and nec essary conditions for the existence of infinitely many positive symmetric entire  solutions which are asymptotic to positive constant multiples of \$ |x|\+\{2n\}(log|x|)\+\{1/(p-1)\}\$ as \$|x|→∞\$. The results can be extended to certain equations of more genera l form, e.g.Δ((Δ\+nu)\+\{p-1*\})=f(|x|, u, |u|,|u\+2u|,\:,|u|\+\{2n-1\})u\+\{-β\}, x∈R^2.

Key words: Nonlinear poly harmonic equation, Positive entire solution, Radial symmetric solution, Singular equation, Fixed point theorem