Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn

Abstract

This paper is concerned with positive solutions of the semilinear polyharmonic equation $(-\Delta)^{m} u = a(x){u}^{\alpha}$ on $\mathbb{R}^{n}$, where $m$ and $n$ are positive integers with $n\gt 2m$, $\alpha\in (-1,1)$. The coefficient a is assumed to satisfy $a(x)\approx{(1+|x|)}^{-\lambda}L(1+|x|)\quad \text{for}\quad x\in \mathbb{R}^{n},$ where $\lambda\in [2m,\infty)$ and $L\in C^{1}([1,\infty))$ is positive with $\frac{tL'(t)}{L(t)}\longrightarrow 0$ as $t\longrightarrow \infty$; if $\lambda=2m$, one also assumes that $\int_{1}^{\infty}t^{-1}L(t)dt\lt \infty$. We prove the existence of a positive solution $u$ such that $u(x)\approx{(1+|x|)}^{-\widetilde{\lambda}}\widetilde{L}(1+|x|) \quad\text{for}\quad x\in \mathbb{R}^{n},$ with $\widetilde{\lambda}:=\min(n-2m,\frac{\lambda-2m}{1-\alpha})$ and a function $\widetilde{L}$, given explicitly in terms of $L$ and satisfying the same condition at infinity. (Given positive functions $f$ and $g$ on $\mathbb{R}^{n}$, $f\approx g$ means that $c^{-1}g\leq f\leq cg$ for some constant $c\gt 1$.)