Positive solutions with specific asymptotic behavior for a polyharmonic problem on $\mathbb{R}^n$

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 > 2m, $\alpha \in (—1,1)$. The coefncient a is assumed to satisfy $a(x) \approx (1+|x|)^{-\lambda}L(1+|x|)$ for $ x \in \mathbb{R}^n$, where $ \lambda \in [2m,\infty)$ and $L \in C^1([q,\infty))$ is positive with $ {tL^\prime (t)}/{L(t)} \rightarrow 0 $ as $ t \rightarrow \infty $ if $ \lambda = 2m$, one also assumes that $\int_1^\infty t^{-1} L(t) dt < \infty $. We prove the existence of a positive solution $u$ such that $ u(x) \approx (1 + |x|)^{-\tilde{\lambda}} L(1+|x|)$ for $ x \in \mathbb{R}^n$, with $\tilde{\lambda} := \text{min}(n-2m, {\lambda-2m}/{1-\alpha})$ and a function $ \tilde{L} $, given explicitly in terms of $L$ and satisfying the same condition as 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 > 1.)