Abstract

This paper deals with the existence and asymptotic analysis of positive continuous solutions for the following nonlinear polyharmonic boundary value problem: \begin{align*} \begin{cases} (-\Delta)^{m}u= b(x)u^{p}, \ \ \text{in} \ D, & \\ \lim\limits_{|x| \rightarrow 1 } \big(|x|^{2}-1\big)^{1-m} u(x) =0, \\ \lim\limits_{|x| \rightarrow \infty } (|x|^{2}-1\big)^{1-m} u(x) = 0. \end{cases} \end{align*} Here $m$ is an integer greater than 2, $p \in(-1,1)$, $ D$ is the complementary of the closed unit ball of $\mathbb{R}^{n} $ with $n >2m$, and the function $b$ is nonnegative and continuous on $D$, satisfying some appropriate assumptions related to Karamata regular variation theory.

Keywords: Polyharmonic equation; positive solutions; Kato class; Karamata classes; Green function; Schauder's fixed point theorem.

MSC: 31B30, 35B09, 35B40, 35J30, 35J40

DOI: 10.57016/MV-MdB8W499

Pages:  1--11