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

Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

Let &Omega; be a bounded domain in [formula] with a smooth boundary [formula]. We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic system [formula] Here r, s &isin; R, &alpha;, &beta; 0 and the functions [formula] are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory